Prescribing the behaviour of geodesics in negative curvature

Presribing the b eha viour of geo desis in negativ e urv ature Jouni P arkk onen F rédéri P aulin No v em b er 19, 2021 Abstrat Giv en a family of (almost) disjoin t stritly on v ex subsets of a omplete negativ ely urv ed Riemannian manifold M , su h as balls, horoballs, tubular neigh b orho o ds of totally geo desi submanifolds, et, the aim of this pap er is to onstrut geo desi ra ys or lines in M whi h ha v e exatly one an exatly presrib ed (big enough) p enetration in one of them, and otherwise a v oid (or do not en ter to o m u h in) them. Sev eral appliations are giv en, inluding a denite impro v emen t of the unlouding problem of [PP1℄, the presription of heigh ts of geo desi lines in a nite v olume su h M , or of spiraling times around a losed geo desi in a losed su h M . W e also pro v e that the Hall ra y phenomenon desrib ed b y Hall in sp eial arithmeti situations and b y S hmidt-Sheingorn for h yp erb oli surfaes is in fat only a negativ e urv ature prop ert y . 1 1 In tro dution The problem of onstruting obstale-a v oiding geo desi ra ys or lines in negativ ely urv ed Riemannian manifolds has b een studied in v arious dieren t on texts. F or example, Dani [Dan ℄ and others [Str , AL, KW℄ ha v e onstruted (man y) geo desi ra ys that are b ounded (i.e. a v oid a neigh b orho o d of innit y) in nonompat Riemannian manifolds. This w ork has deep onnetions with Diophan tine appro ximation problems, see for instane the pap ers b y Sulliv an [Sul ℄, Klein b o  k-Margulis [KM℄ and Hersonsky-P aulin [HP5 ℄. Hill and V elani [HV℄ and others (see for instane [HP3 ℄) ha v e studied the shrinking target problem for the geo desi o w. S hro eder [S hr ℄ and others [BSW℄ ha v e w ork ed on the onstrution of geo desi lines a v oiding giv en subsets, see also the previous w ork [PP1 ℄ of the authors on the onstrution of geo desi ra ys and lines a v oiding a uniformly shrunk family of horoballs. In this pap er, w e are in terested in onstruting geo desi ra ys or lines in negativ ely urv ed Riemannian manifolds whi h, giv en some family of obstales, ha v e exatly one an exatly presrib ed (big enough) p enetration in one of them, and otherwise a v oid (or do not en ter to o m u h in) them. W e also study an asymptoti v ersion of this problem. This in tro dution on tains a sample of our results (see also [PP2 ℄). Giv en a horoball H of en ter ξ or a ball of en ter x and radius r in a CA T ( − 1) metri spae (su h as a omplete simply onneted Riemannian manifold of setional urv ature at most − 1 ), for ev ery t ≥ 0 , let H [ t ] b e the onen tri horoball or ball on tained in H , whose b oundary is at distane t from the b oundary of H (with H [ t ] empt y if H is a ball of radius r and t > r ). The follo wing result (see Setion 4.1 ) greatly impro v es the main 1 AMS o des: 53 C 22, 11 J 06, 52 A 55, 53 D 25. Keyw ords: geo desis, negativ e urv ature, horoballs, Lagrange sp etrum, Hall ra y . 1 results, Theorem 1.1 and Theorem 4.5, of [PP1 ℄. The fat that the onstan t µ 0 is univ ersal (and not v ery big, though not optimal) is indeed remark able. Theorem 1.1 L et X b e a pr op er ge o desi CA T ( − 1) metri sp a e with ar wise  onne te d b oundary ∂ ∞ X and extendible ge o desis, let ( H α ) α ∈ A b e any family of b al ls or hor ob al ls with p airwise disjoint interiors, and let µ 0 = 1 . 53 4 . F or every x in X − S α ∈ A H α , ther e exists a ge o desi r ay starting fr om x and avoiding H α [ µ 0 ] for every α . F rom no w on, w e denote b y M a omplete onneted Riemannian manifold with se- tional urv ature at most − 1 and dimension at least 3 . If M has nite v olume and is non ompat, let e b e an end of M . Let V e b e the maximal Margulis neigh b orho o d of e (see for instane [ BK , Bo w , HP5℄ and Setion 5.1). If ρ e is a minimizing geo desi ra y in M starting from a p oin t in the b oundary of V e and on v erging to e , let h t e : M → R b e the heigh t map dened b y h t e ( x ) = lim t →∞ ( t − d ( ρ e ( t ) , x )) . The maximum height sp e trum MaxSp( M , e ) of the pair ( M , e ) is the subset of ] − ∞ , + ∞ ] onsisting of elemen ts of the form sup t ∈ R h t e ( γ ( t )) where γ is a lo ally geo desi line in M . As a onsequene of Theorem 1.1 (see Corollary 4.4 ), w e pro v e that in an y (nonom- pat) nite v olume omplete Riemannian manifold with dimension at least 2 and setional urv ature at most − 1 , there exists univ ersally lo w losed geo desis. F urthermore, w e ha v e the follo wing result on the upp er part of the sp etrum: Theorem 1.2 If M has nite volume and e is an end of M , then MaxSp( M , e )  ontains the interval [4 . 2 , + ∞ ] . F or more preise analogous statemen ts when M is geometrially nite, and for nite subsets of usps of M , see Setion 5.1. S hmidt and Sheingorn [SS ℄ pro v ed the t w o- dimensional analog of Theorem 1.2 in onstan t urv ature − 1 . They sho w ed that the maxim um heigh t sp etrum of a nite area h yp erb oli surfae with resp et to an y usp on tains the in terv al [4 . 61 , + ∞ ] . The previous result is obtained b y studying the p enetration prop erties of geo desi lines in a family of horoballs. Our next theorem onerns families of balls (see Setion 5.1 for generalizations). Theorem 1.3 L et x b e p oint in M with r = inj M x ≥ 56 . Then, for every d ∈ [2 , r − 54] , ther e exists a lo  al ly ge o desi line γ p assing at distan e exatly d fr om x at time 0 and r emaining at distan e gr e ater than d fr om x at any nonzer o time. Giv en a losed geo desi L in M , the b eha vior of a lo ally geo desi ra y γ in M with resp et to L is t ypially that γ spirals around L for some time, then w anders a w a y from L , then spirals again for some time around L , then w anders a w a y , et. Our next aim is to onstrut su h a γ whi h has exatly one (big enough) exatly presrib ed spiraling length, and all of whose other spiraling lengths are b ounded ab o v e b y some uniform onstan t. Let us mak e this preise (see the gure in Setion 5.2 ). Let L b e an em b edded ompat totally geo desi submanifold in M with 1 ≤ dim L ≤ dim M − 1 , and ǫ > 0 small enough so that the (losed) ǫ -neigh b orho o d N ǫ L of L is a tubular neigh b orho o d. F or ev ery lo ally geo desi line γ in M , the set of t ∈ R su h that γ ( t ) b elongs to N ǫ L is the disjoin t union of maximal losed in terv als [ s n , t n ] , with 2 s n ≤ t n < s n +1 . F or ev ery su h in terv al, let x n (resp. y n ) b e the origin of the lo ally geo desi ra y starting from a p oin t of L p erp endiularly to L , whi h is prop erly homotopi, b y a homotop y with origins in L , to the pieewise geo desi path starting with the geo desi segmen t from the losest p oin t of γ ( s n ) (resp. γ ( t n ) ) on L to γ ( s n ) (resp. γ ( t n ) ), and then follo wing γ at times less than s n (resp. more than t n ). The length of the lo ally geo desi path in L b et w een x n , y n in the ob vious homotop y lass will b e alled a fel low-tr aveling time of γ along L (see Setion 5.2 ). Theorem 1.4 L et L b e as ab ove. Ther e exist  onstants c, c ′ > 0 , dep ending only on ǫ , suh that for every h ≥ c , ther e exists a lo  al ly ge o desi line in M , having one fel low-tr aveling time exatly h , al l others b eing at most c ′ . See Setion 5.2 for an extension of Theorem 1.4 when L is not neessarily em b edded, and to nitely man y disjoin t su h neigh b orho o ds N ǫ L . If M has nite v olume, w e also onstrut b ounded lo ally geo desi lines with the ab o v e prop ert y (with a on trol of the heigh ts uniform in ǫ ). In onstan t urv ature, w e an also presrib e one of the p enetration lengths | t n − s n | at least c , while k eeping all the other ones at most c ′ . S hmidt and Sheingorn [SS ℄ sk et h the pro of of a result for h yp erb oli surfaes whi h is analogous to Theorem 1.4 with a dieren t w a y of measuring the anit y of lo ally geo desi lines. Other results ab out the spiraling prop erties of geo desi lines around losed geo desis will b e giv en in [HP6, HPP ℄. F or our next result, w e sp eialize to the ase where M is a h yp erb oli 3 -manifold. See Setion 5.3 for a more general statemen t, and for instane [ MT℄ for referenes on 3 -manifolds and Kleinian groups. Theorem 1.5 L et N b e a  omp at,  onne te d, orientable, irr e duible, aylindri al, ator oi- dal, b oundary in ompr essible 3 -manifold with b oundary, with ∂ N having exatly one torus  omp onent e . F or every  omp at subset K in the sp a e G F ( N , e ) of (isotopy lasses of )  omplete ge ometri al ly nite hyp erb oli metris in the interior of N with one usp, ther e exists a  onstant c ≥ 0 suh that for every h ≥ c and every σ ∈ K , ther e exists a lo  al ly ge o desi line γ  ontaine d in the  onvex  or e of σ suh that the maximum height of γ is exatly h . If M has nite v olume and e is an end of M , dene the asymptoti height sp e trum LimsupS p( M , e ) of the pair ( M , e ) to b e the subset of ] − ∞ , + ∞ ] onsisting of elemen ts of the form lim sup t ∈ R h t e ( γ ( t )) where γ is a lo ally geo desi line in M . Theorem 1.6 (The ubiquit y of Hall ra ys) If M has nite volume and e is an end of M , then LimsupS p( M , e )  ontains [6 . 8 , ∞ ] . The in terv al giv en b y Theorem 1.6 is alled a Hal l r ay . Note that the v alue 6 . 8 is uniform on all ouples ( M , e ) , but w e do not kno w the optimal v alue. If M is the one-ended h yp erb oli 2 -orbifold PSL 2 ( Z ) \ H 2 R where H 2 R is the real h yp erb oli plane with setional urv ature − 1 , then the existene of a Hall ra y follo ws from the w ork of Hall [Hal1, Hal2℄ on on tin ued frations. F reiman [F re ℄ (see also [Slo℄) has determined the maximal Hall ra y of PSL 2 ( Z ) \ H 2 R , whi h is appro ximately [3 . 02 , + ∞ ] . The generalit y of Theorem 1.6 pro v es in partiular that the Hall ra y phenomenon is neither an arithmeti nor a onstan t urv ature prop ert y . See Setion 5.4 for a more preise v ersion of Theorem 1.6 , whi h is v alid also in the geometrially nite ase. 3 The results of Hall and F reiman ited ab o v e w ere originally form ulated in terms of Diophan tine appro ximation of real n um b ers b y rationals. The pro jetiv e ation of the mo dular group PSL 2 ( Z ) on the upp er halfplane pro vides a w a y to obtain the geometri in terpretation. W e onlude this sample of our results b y giving appliations of our metho ds to Diophan tine appro ximation problems (see Setion 6 for generalizations in the framew ork of Diophan tine appro ximation on negativ ely urv ed manifolds, dev elopp ed in [HP3, HP4, HP5 ℄). These results w ere announed in [ PP2 ℄. Theorem 1.7 L et m b e a squar efr e e p ositive inte ger, and I a non-zer o ide al in an or der in the ring of inte gers − m of the imaginary quadr ati numb er eld Q ( i √ m ) . F or every x ∈ C − Q ( i √ d ) , let c ( x ) = lim inf ( p,q ) ∈× I , h p,q i = , | q |→∞ | q | 2    x − p q    b e the appr oximation  onstant of the  omplex numb er x by elements of I − 1 , and Sp Lag the L agr ange sp e trum  onsisting of the r e al numb ers of the form c ( x ) for some x ∈ C − Q ( i √ m ) . Then Sp Lag  ontains the interval [0 , 0 . 033 ] . Theorem 1.7 follo ws from Hall's result and from the w ork of P oitou [P oi ℄ in the par- tiular ase I = = − m . Other arithmeti appliations of our geometri metho ds an b e obtained b y v arying the (non uniform) arithmeti lattie in the isometry group of a negativ ely urv ed symmetri spae. W e only state the follo wing result in this in tro dution (with the notation of Setion 6.1), see Setion 6.4 and [PP2 ℄ for other ones. Theorem 1.8 L et Q ( R ) b e the r e al quadri { ( z , w ) ∈ C 2 : 2 Re z − | w | 2 = 0 } endowe d with the Lie gr oup law ( z , w ) · ( z ′ , w ′ ) = ( z + z ′ + w ′ w , w + w ′ ) and Q ( Q ) = Q ( R ) ∩ Q ( i ) 2 b e its r ational p oints. If r = ( p/q , p ′ /q ) ∈ Q ( Q ) with p, p ′ , q ∈ Z [ i ] r elatively prime, let h ( r ) = | q | . L et d ′ Cyg b e the left-invariant distan e on Q ( R ) suh that d ′ Cyg (( z , w ) , (0 , 0)) = p 2 | z | + | w | 2 . F or every x ∈ Q ( R ) − Q ( Q ) , let c ( x ) = lim inf r ∈ Q ( Q ) , h ( r ) →∞ h ( r ) d ′ Cyg ( x, r ) b e the appr oximation  onstant of x by r ational p oints, and Sp Lag the L agr ange sp e trum  onsisting of the r e al numb ers of the form c ( x ) for some x ∈ Q ( R ) − Q ( Q ) . Then Sp Lag  ontains the interval [0 , 0 . 047 ] . The pap er is organized as follo ws. In Setion 2, w e dene a lass of uniformely stritly on v ex subsets of metri spaes, that w e all ǫ - onvex subsets . W e study the in teration of geo desi ra ys and lines with ǫ -on v ex sets in CA T( − 1 )-spaes. In partiular, w e giv e v arious estimates on the distane b et w een the en tering and exiting p oin ts in an ǫ -on v ex set of t w o geo desi ra ys starting from a xed p oin t in the spae and of t w o geo desi lines starting from a xed p oin t in the b oundary at innit y . Setion 3 is dev oted to dening and studying sev eral p enetration maps whi h are used to measure the p enetration of geo desi ra ys and lines in an ǫ -on v ex set. W e emphasize the ase of p enetration maps in horoballs, balls and tubular neigh b orho o ds of totally geo desi submanifolds. W e sho w that in a n um b er of geometrially in teresting ases, it is p ossible to adjust the p enetration of a geo desi line or ra y in one ǫ -on v ex set while k eeping the p enetration in another set xed. Setion 4 on tains the indutiv e onstrution that giv es geo desi ra ys and lines with presrib ed 4 maximal p enetration with resp et to a giv en olletion of ǫ -on v ex sets. As a w arm-up for the onstrution, w e pro v e Theorem 1.1 in Subsetion 4.1 . The other theorems in the in tro dution b esides the last t w o and a n um b er of others are pro v ed in Setion 5 where the results of Setion 4 are applied in the ases studied in Setion 3. Finally , w e giv e our arithmeti appliations in Setion 6. A know le dgments. Ea h author a kno wledges the supp ort of the other author's institution, where part of this w ork w as done. This resear h w as supp orted b y the Cen ter of Exellene "Geometri analysis and mathematial ph ysis" of the A adem y of Finland. W e thank P . P ansu, Y. Bugeaud, A. S hmidt, P . Gilles, A. Guilloux, D. Harari for v arious disussions and ommen ts on this pap er. 2 On strit on v exit y in CA T ( − 1) spaes 2.1 Notations and ba kground In this setion, w e in tro due some of the ob jets whi h are en tral in this pap er. W e refer to [BH , GH ℄ for the denitions and basi prop erties of CA T( − 1 ) spaes. Our referene for h yp erb oli geometry is [Bea ℄. Let ( X, d ) b e a prop er geo desi CA T ( − 1) metri spae, and X ∪ ∂ ∞ X b e its om- patiation b y the asymptoti lasses of geo desi ra ys. By a ge o desi line (resp. r ay or se gment ) in X , w e mean an isometri map γ : R → X (resp. γ : [ ι 0 , + ∞ [ → X with ι 0 ∈ R or γ : [ a, b ] → X , with a ≤ b ). W e sometimes also denote b y γ the image of this map. F or x, y in X , w e denote b y [ x, y ] the (unique) losed geo desi segmen t b et w een x, y , with the ob vious extension to op en and half-op en geo desi segmen ts, ra ys and lines (with one or t w o endp oin ts in ∂ ∞ X ). W e sa y that X has extendible ge o desis if ev ery geo desi segmen t an b e extended to a geo desi line. W e denote b y T 1 X the spae of geo desi lines in X , endo w ed with the ompat-op en top ology . When X is a Riemannian manifold, the spae T 1 X oinides with the usual denition of the unit tangen t bundle, up on iden tifying a geo desi line γ and its (unit) tangen t v etor ˙ γ (0) at time t = 0 . F or ev ery geo desi ra y or line γ , w e denote b y γ (+ ∞ ) the p oin t of ∂ ∞ X to whi h γ ( t ) on v erges as t → + ∞ , and w e dene γ ( − ∞ ) similarly when γ is a geo desi line. W e sa y that a geo desi line (resp. ra y) γ starts from a p oin t ξ ∈ ∂ ∞ X (resp. ξ ∈ X ) if ξ = γ ( −∞ ) (resp. γ ( ι 0 ) = ξ ). F or ev ery ξ in X ∪ ∂ ∞ X , w e denote b y T 1 ξ X the spae of geo desi lines (if ξ ∈ ∂ ∞ X ) or ra ys (if ξ ∈ X ) starting from ξ , endo w ed with the ompat-op en top ology . If Y is a subset of X and ξ a p oin t in X ∪ ∂ ∞ X , the shadow of Y se en fr om ξ is the set ξ Y of p oin ts γ (+ ∞ ) where γ is a geo desi ra y or line starting from ξ and meeting Y . The Busemann funtion β ξ : X × X → R at a p oin t ξ in ∂ ∞ X is dened b y β ξ ( x, y ) = lim t → + ∞  d ( x, ρ ( t )) − d ( y , ρ ( t ))  , where ρ is an y geo desi ra y ending at ξ . The funtion x 7→ β ξ ( x, y ) an b e though t of as a normalized signed distane to ξ ∈ ∂ ∞ X , or as the height of the p oin t x with resp et to ξ (relativ e to y ). A ordingly , if β ξ ( x, y ) = β ξ ( x ′ , y ) , then the p oin ts x and x ′ are said to b e e quidistant to ξ . If ξ ∈ X , w e dene β ξ ( x, y ) = d ( x, ξ ) − d ( y , ξ ) . This is on v enien t in Setion 4.2 and in the pro of of Corollary 5.5. F or ev ery x, y , z in X and ξ ∈ X ∪ ∂ ∞ X , w e ha v e β ξ ( x, y ) + β ξ ( y , z ) = β ξ ( x, z ) , 5 β ξ ( x, x ) = 0 , and | β ξ ( x, y ) | ≤ d ( x, y ) . A hor ob al l in X en tered at ξ ∈ ∂ ∞ X is the preimage of [ s, + ∞ [ for some s in R b y the map y 7→ β ξ ( x, y ) for some x in X . If H = { y ∈ X : β ξ ( x, y ) ≥ s } is a horoball, w e dene its b oundary hor ospher e b y ∂ H = { y ∈ X : β ξ ( x, y ) = s } , and for ev ery t ≥ 0 , its t -shrunk hor ob al l b y H [ t ] = { y ∈ X : β ξ ( x, y ) ≥ s + t } . (In [PP1 ℄, w e denoted H [ t ] b y H ( t ) .) Similarly , if B is a ball of en ter x and radius r , for ev ery t ≤ r , w e denote b y B [ t ] the ball of en ter x and radius r − t . By on v en tion, if t > r , dene B [ t ] = ∅ . Note that for ev ery ball or horoball H , w e ha v e H [ t ′ ] ⊂ H [ t ] if t ′ ≥ t . The p oin t at innit y of an horoball H is denoted b y H [ ∞ ] . Note that, in this pap er, all balls and horoballs in X are assumed to b e losed. Reall that a subset C in a CA T ( − 1) metri spae is  onvex if C on tains the geo desi segmen t b et w een an y t w o p oin ts in C . Let C b e a on v ex subset in X . W e denote b y ∂ ∞ C its set of p oin ts at innit y , and b y ∂ C its b oundary in X . If C is nonempt y and losed, for ev ery ξ in ∂ ∞ X , w e dene the losest p oint to ξ on the  onvex set C to b e the follo wing p oin t p in C ∪ ∂ ∞ C : if ξ / ∈ ∂ ∞ C , then p b elongs to C and maximizes the map y 7→ β ξ ( x 0 , y ) for some (hene an y) giv en p oin t x 0 in X ; if ξ ∈ ∂ ∞ C , then w e dene p = ξ . This p exists, is unique, and dep ends on tin uously on ξ , b y the prop erties of CA T ( − 1) -spaes. If x, y , z ∈ X ∪ ∂ ∞ X , w e denote b y ( x, y , z ) the triangle formed b y the three geo desi segmen ts, ra ys or lines with endp oin ts in { x, y , z } . Reall that if α : t 7→ α t and β : t 7→ β t are t w o (germs of ) geo desi segmen ts starting from a p oin t x 0 in X at time t = 0 , if ( x 0 , α t , β t ) for t > 0 small enough is a omparison triangle to ( x 0 , α t , β t ) in the real h yp erb oli plane H 2 R , then the  omp arison angle b et w een α and β at x 0 is the limit, whi h exists, of the angle ∠ x 0 ( α t , β t ) as t tends to 0 . W e end this setion with the follo wing (w ell kno wn) exerises in h yp erb oli geometry . Lemma 2.1 F or al l p oints x, y in X and z in X ∪ ∂ X , and every t in [0 , d ( x, z )] , if x t is the p oint on [ x, z ] at distan e t fr om x , then d ( x t , [ y , z ]) ≤ e − t sinh d ( x, y ) ≤ 1 2 e − t + d ( x,y ) . Pro of. By omparison, w e ma y assume that X = H 2 R . As it do es not derease d ( x t , [ y , z ]) to replae z b y the p oin t at innit y of the geo desi ra y starting from x and passing through z , w e ma y assume that z is the p oin t at innit y in the upp er halfspae mo del of H 2 R . Let p b e the orthogonal pro jetion of x t on the geo desi line γ through y and z . Assume rst that p b elongs to [ y , z [ . 6 If w e replae y b y the orthogonal pro jetion of x on γ , then w e derease d ( x, y ) , and do not  hange t and d ( x t , [ y , z ]) . Hene w e ma y assume that y = i and x is on the (Eulidean) irle of en- ter 0 and radius 1 . If α is the (Eulidean) angle at 0 b et w een the horizon tal axis and the (Eulidean) line from 0 passing through x , then an easy omputation in h yp erb oli geometry (see also [Bea ℄, page 145) giv es sinh d ( x, y ) = cos α/ sin α . Similarly , sinh d ( x t , [ y , z ]) = cos α/ ( e t sin α ) . So that d ( x t , [ y , z ]) ≤ sinh d ( x t , [ y , z ]) = e − t sinh d ( x, y ) . cos α sin α α e t sin α x t z x y p Assume no w that p do es not b elong to [ y , z [ . In partiular, y 6 = x t . Let x t b e the p oin t at same distane from y as x t (and on the same side) su h that y is the orthogonal pro jetion of x t on γ , so that d ( x t , [ y , z ]) = d ( x t , y ) = d ( x t , y ) = d ( x t , [ y , z ]) . Let x b e the in tersetion of the geo desi line from z through x t with the (h yp erb oli) irle of en ter y and radius d ( x, y ) , so that d ( x, y ) = d ( x, y ) . Then, with t = d ( x t , x ) , w e ha v e t ≥ t , as the angle at x t of [ x t , x ] with the outgoing unit v etor of the geo desi ra y from y through x t is bigger than the orresp onding one for x t and x . Hene w e ma y assume that x t = x t and x = x . As then the orthogonal pro jetion of x t on the geo desi line through y and z is y , this redues the situation to the rst ase treated ab o v e.  x t x y p x x t Lemma 2.2 F or every ǫ > 0 , if c 0 ( ǫ ) = 2 log  2(1 + e ǫ/ 2 ) sinh ǫ ǫ  , (- 1 -) then for al l p oints a, b, a ′ , b ′ in X suh that d ( a, a ′ ) ≤ ǫ , d ( b, b ′ ) ≤ ǫ , d ( a, b ) ≥ c 0 ( ǫ ) , if m is the midp oint of the ge o desi se gment [ a, b ] , then d ( m, [ a ′ , b ′ ]) ≤ ǫ 2 . Pro of. Let p b e the p oin t in [ a, b ′ ] the losest to m , and q the p oin t of [ a ′ , b ′ ] the losest to p . Let t = d ( a, m ) = d ( b, m ) = d ( a, b ) / 2 . By Lemma 2.1, w e ha v e d ( m, p ) ≤ e − d ( b,m ) sinh d ( b, b ′ ) ≤ e − t sinh ǫ and, as d ( m, p ) ≤ ǫ/ 2 b y on v exit y , d ( p, q ) ≤ e − d ( a,p ) sinh d ( a, a ′ ) ≤ e − d ( a,m )+ d ( m ,p ) sinh d ( a, a ′ ) ≤ e − t + ǫ/ 2 sinh ǫ . Hene d ( m, q ) ≤ d ( m, p ) + d ( p, q ) ≤ e − t (1 + e ǫ/ 2 ) sinh ǫ , and the result follo ws b y the assumption on d ( a, b ) .  Remark. If w e w an t a simpler expression, w e an also tak e c 0 ( ǫ ) = 3 ǫ + 4 log 2 . 7 2.2 En tering and exiting ǫ -on v ex subsets F or ev ery subset A in X and ǫ > 0 , w e denote b y N ǫ A the losed ǫ -neigh b orho o d of A in X . F or ev ery ǫ > 0 , a subset C of X will b e alled ǫ - onvex if there exists a on v ex subset C ′ in X su h that C = N ǫ C ′ . As the metri spae X is CA T ( − 1) , it is easy to see that an ǫ -on v ex subset C is losed, on v ex, equal to the losure of its in terior, and stritly  onvex in the sense that for ev ery geo desi line γ meeting C in at least t w o p oin ts, the segmen t γ ∩ C is the losure of γ ∩ ◦ C . If X is a smo oth Riemannian manifold, then an ǫ -on v ex subset has a C 1 , 1 -smo oth b oundary , see [W al ℄. Examples. (1) F or ev ery ǫ > 0 , an y ball of radius at least ǫ is ǫ -on v ex, and an y horoball is ǫ -onv ex. Con v ersely , as pro v ed b elo w, if a subset C ⊂ X is ǫ -on v ex for ev ery ǫ > 0 , then C is X , ∅ or a horoball. A ordingly , w e will sometimes refer to horoballs as ∞ -on v ex subsets. T o pro v e the ab o v e statemen t, assume that C 6 = X , ∅ and that for all ǫ > 0 , there exists a on v ex subset C − ǫ in X su h that C = N ǫ C − ǫ . F or ev ery x in ∂ C (note that ∂ C is non empt y as C 6 = X , ∅ ) and ev ery t ≥ 0 , let x t b e the p oin t of the losed on v ex subset C − t whi h is the losest to x . Then t 7→ x t is a geo desi ra y , whi h on v erges to a p oin t alled x ∞ . W e laim that x ∞ = y ∞ for ev ery x, y in ∂ C . Otherwise, the geo desi segmen t b et w een x t and y t , on tained in C − t b y on v exit y , on v erges to the geo desi line b et w een x ∞ = y ∞ . Hene, the p oin t x t w ould not b e the losest one to x , for t big enough. Therefore ∂ C is an horosphere whose p oin t at innit y is x ∞ , and b y on v exit y , C is an horoball. (2) F or manifolds, the prop ert y for a losed on v ex subset with nonempt y in terior to b e ǫ -on v ex is related with extrinsi urv ature prop erties of its b oundary . Let us explain this relationship. Let ( X, h· , ·i ) b e a omplete simply onneted smo oth Riemannian manifold of dimen- sion m ≥ 3 with pin hed negativ e setional urv ature − b 2 ≤ K ≤ − a 2 < 0 . Let C b e a ompat on v ex subset of X with nonempt y in terior, and with C ∞ -smo oth b ound- ary S = ∂ C . (Compatness and C ∞ instead of C 1 , 1 are not really neessary , but as statemen ts and pro ofs are then simpler, w e will w ork only under these h yp otheses). Let I I S : T S ⊕ T S → R b e the seond fundamen tal form of S asso iated to the in w ard normal unit v etor eld ~ n along S , that is I I S ( V , W ) = h∇ V W , ~ n i = −h∇ V ~ n, W i , where V , W are tangen t v etors to S at the same p oin t, extended to v etor elds tangen t to S at ev ery p oin t of S (the denition of I I S dep ends on the  hoie b et w een ~ n and − ~ n , and the v arious referenes dier on that p oin t, see for instane [GHL, page 217℄, [P et , page 36℄ vs [Gra , page 37℄; w e ha v e  hosen the in w ard p oin ting v etor eld in order for the symmetri bilinear form I I S to b e nonnegativ e, b y on v exit y of C ). Let I I S b e the upp er b ound of I I S ( V , V ) for ev ery unit tangen t v etor V to S . Prop osition 2.3 • If I I S ≤ a coth ( aǫ ) , then C is ǫ - onvex. • If C is ǫ - onvex, then I I S ≤ b coth ( bǫ ) . Pro of. Assume rst that C is ǫ -on v ex. Let x b e a p oin t in S and y = exp x ( ǫ ~ n ( x )) . Note that the sphere S X ( y , ǫ ) of en ter y and radius ǫ in X is on tained in C , as C is ǫ -on v ex. 8 Lo ally o v er the tangen t spae T x S = T x ( S X ( y , ǫ )) , the graph of S X ( y , ǫ ) is ab o v e the graph of S (when ~ n p oin ts up w ards). Hene, for ev ery V in T x S , w e ha v e I I S ( V , V ) ≤ I I S X ( y, ǫ ) ( V , V ) . As the setional urv ature of X is at least − b 2 , w e ha v e b y omparison I I S X ( y, ǫ ) ≤ b coth( bǫ ) (see for instane [P et , page 145℄). The seond result follo ws. Note that b coth( bǫ ) is the (v alue on unit tangen t v etors of ) the seond fundamen tal form (with resp et to the in w ard normal unit v etor eld) of a sphere of radius ǫ in the real h yp erb oli 3 -spae with onstan t urv ature − b 2 ). Assume no w that I I S ≤ a coth ( aǫ ) . F or ev ery t ≥ 0 and x in S , let x t = exp x ( t ~ n ( x )) . Iden tify b y parallel transp ort | | x x t : T x t X → T x X along t 7→ x t the tangen t spaes T x t X with T x X . F or ev ery x in S , let A ( t ) and R ( t ) b e the symmetri endomorphisms of T x S dened b y v 7→ −| | x x t  ∇ | | x t x ( v ) ˙ x t  and v 7→ R ( | | x t x ( v ) , ˙ x t ) ˙ x t , resp etiv ely , where R is the urv ature tensor of X . F or ev ery v ∈ T 1 S , note that h R ( t ) v , v i , b eing the urv ature of the plane generated b y the orthonormal tangen t v etors || x t x v and ˙ x t at x t , is at most − a 2 . It is w ell kno wn that t 7→ A ( t ) satises the follo wing matrix Riati equation ˙ A ( t ) + A ( t ) 2 + R ( t ) = 0 (see for instane Theorem 3.6 on page 37 of [P et ℄). The follo wing standard result, on- trolling the time of explosion of the solution to the matrix Riati equation, follo ws for instane from (a minor mo diation of ) Theorem 2.3 on page 144 of [P et ℄. Lemma 2.4 L et t 7→ A ( t ) b e a smo oth map fr om a neighb orho o d of 0 in R to the sp a e of symmetri matri es on an Eulide an sp a e, suh that ˙ A ( t ) + A ( t ) 2 − a 2 Id is nonne gative, and the biggest eigenvalue of A (0) is at most a coth aǫ . Then t 7→ A ( t ) is dene d and smo oth at le ast on ] − ǫ, + ǫ [ , and A ( t ) is nonne gative for every t in ] − ǫ, + ǫ [ .  As h A ( t ) v , v i = I I S ( v , v ) for ev ery v in T 1 S and I I S ≤ a coth( aǫ ) , it follo ws that for ev ery t ∈ [0 , ǫ [ , the map x 7→ x t is a smo oth immersion of the ompat ( m − 1) -manifold S in to X , su h that, if S t is the image, then I I S t is ev erywhere nonnegativ e. As m ≥ 3 , it follo ws from the Hadamard-Alexander theorem (see for instane [ Ale℄) that S t b ounds a on v ex subset C t , with C t ⊂ C t ′ if t ′ ≤ t . If C ′ = ∩ t ∈ [0 ,ǫ [ C t , then C ′ is ompat, nonempt y , on v ex, and C = N ǫ C ′ , so that C is ǫ -on v ex.  In partiular, the ab o v e result implies that if M has onstan t urv ature − a 2 , then C is ǫ -on v ex in our sense if and only if I I S ≤ a coth ( aǫ ) . Our notion of ǫ -on v exit y is hene related to, but dieren t from, the opp osite of the notion of λ -on v exit y studied for instane in [GR, BGR, BM ℄. Alexander and Bishop [AB℄ (see also [Lyt ℄), ha v e in tro dued a natural notion of an extrinsi urv ature b ounded from ab o v e for subspaes of CA T( κ ) -spaes, extending the notion of ha ving a b ounded (absolute v alue of the) seond fundamen tal form for submanifolds of Riemannian manifolds. Th us, the opp osite of this onept of [AB℄ is related to our notion of ǫ -on v ex subsets (see in partiular Prop osition 6.1 in [AB℄). The rest of this setion is dev oted to sev eral lemmas onerning the relativ e distanes b et w een en tering p oin ts and exiting p oin ts, in and out of an ǫ -on v ex subset of X , of t w o geo desi ra ys or lines starting from the same p oin t. The asymptoti b eha viour of the v arious onstan ts app earing in this setion is desrib ed in Remark 2.9 . 9 Lemma 2.5 L et C b e a  onvex subset in X , let ǫ > 0 and let ξ 0 ∈ ( X ∪ ∂ ∞ X ) − ( N ǫ C ∪ ∂ ∞ C ) . If two ge o desi se gments, r ays or lines γ , γ ′ whih start fr om ξ 0 interse t N ǫ C , then the rst interse tion p oints x, x ′ of γ , γ ′ r esp e tively with N ǫ C ar e at a distan e at most c ′ 1 ( ǫ ) = 2 arsinh(coth ǫ ) . Pro of. Let y and y ′ b e the losest p oin ts in C to x and x ′ resp etiv ely . As x, x ′ ∈ ∂ N ǫ [ y , y ′ ] , it is suien t to pro v e the result when C = [ y , y ′ ] . W e ma y assume that x 6 = x ′ , and, b y a on tin uit y argumen t, that y 6 = y ′ . Let us onstrut a p en tagon in H 2 R with v er- ties ξ 0 , x, y , x ′ , y ′ b y gluing together the om- parison triangles of ( ξ 0 , x, x ′ ) , ( x, x ′ , y ′ ) and ( x, y ′ , y ) . By omparison (see for instane [BH, Prop. 1.7.(4)℄), the omparison angles at x, y , x ′ , y ′ are at least π / 2 . Hene, the seg- men ts or ra ys ] ξ 0 , x [ and ] ξ 0 , x ′ [ do not meet N ǫ [ y , y ′ ] , and the p oin t y is the losest p oin t on [ y , y ′ ] to x . x y y ′ x ′ γ ′ ξ 0 ǫ γ ǫ F urthermore, y ′ is the losest p oin t on [ y , y ′ ] to x ′ . Indeed, the angle at y ′ of the p en tagon is at most 3 π / 2 sine ∠ y ′ ( y , x ) ≤ π / 2 and ∠ y ′ ( x, x ′ ) ≤ π . Therefore, if b y absurd z ∈ [ y , y ′ [ is losest to x ′ , the geo desi segmen t [ x ′ , z ] in tersets [ y ′ , x ] at a p oin t u . If z ∈ [ y ′ , y ] and u ∈ [ y ′ , x ] are su h that d ( y ′ , z ) = d ( y ′ , z ) and d ( y ′ , u ) = d ( y ′ , u ) , then b y omparison d ( x ′ , z ) ≤ d ( x ′ , u ) + d ( u, z ) ≤ d ( x ′ , u ) + d ( u, z ) = d ( x ′ , z ) < d ( x ′ , y ′ ) = d ( x ′ , y ′ ) , a on tradition. As d ( x, x ′ ) = d ( x, x ′ ) , w e only ha v e to pro v e that d ( x, x ′ ) ≤ c ′ 1 ( ǫ ) , i.e. w e ma y assume that X = H 2 R . Up to replaing ξ 0 b y the p oin t at innit y of the geo desi ra y starting at x and passing through ξ 0 , w e ma y assume that ξ 0 is at innit y . By homogeneit y , w e ma y assume that ξ 0 is the p oin t at innit y ∞ in the upp er halfplane mo del of H 2 R . As a geo desi line starting from ∞ and meeting the ǫ -neigh b orho o d of a v ertial geo desi segmen t en ters it in the sphere of radius ǫ en tered at its highest p oin t, w e ma y assume that C is a segmen t of the geo desi line ℓ b et w een the p oin ts − 1 and 1 of the real line. As there are geo desi lines starting at ∞ whose rst in tersetion p oin ts with N ǫ ℓ are at distane at least d ( x, x ′ ) , w e ma y assume that C = ℓ . (cosh ǫ, sinh ǫ ) ( − cosh ǫ , sinh ǫ ) ℓ (0 , sinh ǫ ) 10 The distane d ( x, x ′ ) is then maximized when the geo desi lines are tangen t to N ǫ ℓ on b oth sides (see the ab o v e gure). The upp er omp onen t of the b oundary of N ǫ ℓ is the in tersetion with H 2 R of the Eulidean irle through ( ± 1 , 0) and (0 , e ǫ ) , hene en tered at (0 , sinh ǫ ) . Th us, w e ma y assume that the p oin ts x, x ′ are ( ± cosh ǫ, sin h ǫ ) . A omputation then yields the result.  The follo wing te hnial result will b e used in Lemma 2.7. Dene, for ev ery ǫ > 0 , c ′′ ( ǫ ) = 2 ǫ arcosh(2 cosh( ǫ/ 2)) . Lemma 2.6 F or every ǫ > 0 , for every  onvex subset C in X , for every a, b in N ǫ C and for every a 0 in [ a, b ] , if d ( a, b ) ≥ c 0 ( ǫ ) and η = 1 c ′′ ( ǫ ) min { d ( a 0 , a ) , d ( a 0 , b ) } ≤ ǫ 2 , then d ( a 0 , C ) ≤ ǫ − η . Pro of. Let ǫ > 0 . Let C, a, b, a 0 , η b e as in the statemen t, and let us pro v e that d ( a 0 , C ) ≤ ǫ − η . By an easy omputation, w e ha v e c ′′ ( ǫ ) ǫ ≤ c 0 ( ǫ ) . By symmetry , w e ma y assume that d ( a, a 0 ) ≤ d ( b, a 0 ) , so that our assumptions giv e the follo wing inequalities: d ( a, a 0 ) = c ′′ ( ǫ ) η ≤ c ′′ ( ǫ ) ǫ/ 2 ≤ c 0 ( ǫ ) / 2 ≤ d ( a, b ) / 2 . (- 2 -) Let a ′ , b ′ b e the p oin ts in C the losest to a, b resp etiv ely . As [ a ′ , b ′ ] is on tained in C , w e ma y assume that C = [ a ′ , b ′ ] . Let m b e the midp oin t of [ a, b ] , and m ′ its losest p oin t on [ a ′ , b ′ ] . By Lemma 2.2, w e ha v e d ( m, m ′ ) ≤ ǫ 2 . As η ≤ ǫ/ 2 , if d ( a, a ′ ) ≤ ǫ − η , then ev ery p oin t in [ a, m ] is at distane at most ǫ − η from C . In partiular, this is true for a 0 , sine d ( a, a 0 ) ≤ d ( a, m ) . Hene, w e ma y assume that d ( a, a ′ ) > ǫ − η . Consider the quadruple ( a, a ′ , m, m ′ ) of p oin ts of X , whi h satises • ǫ − η < d ( a, a ′ ) ≤ ǫ , • d ( m, m ′ ) ≤ ǫ 2 , • a ′ is the p oin t in [ a ′ , m ′ ] the losest to a , and • m ′ is the p oin t in [ a ′ , m ′ ] the losest to m . Dene t = t ( a, a ′ , m, m ′ ) as the distane b et w een a and the p oin t z = z ( a, a ′ , m, m ′ ) in [ a, m ] at distane ǫ − η from [ a ′ , m ′ ] (whi h exists and is unique b y on v exit y), see the gure b elo w. m ′ L ′ m ∗ m ′ ∗ m a ∗∗∗ a ∗∗ z a ′ ∗ ǫ ǫ − η a ′ a a ∗ z ∗ 11 W e laim that t ≤ c ′′ ( ǫ ) η = d ( a, a 0 ) . Before pro ving this laim, w e note that it implies b y Equation (- 2 - ) that t ≤ d ( a, a 0 ) ≤ d ( a, b ) / 2 , hene, b y on v exit y , d ( a 0 , [ a ′ , m ′ ]) ≤ ǫ − η , and Lemma 2.6 will follo w. W e will mak e sev eral redutions, in order to rea h a situation where easy omputations will b e p ossible. First w e ma y assume, b y omparison, that X = H 2 R . By an appro ximation argumen t, w e ma y assume that a ′ 6 = m ′ 6 = m . The assumptions on the quadruple ( a, a ′ , m, m ′ ) imply that the angles ∠ a ′ ( a, m ′ ) and ∠ m ′ ( m, a ′ ) are at least π 2 . If the segmen t [ a, m ] uts the segmen t [ a ′ , b ′ ] in a p oin t u , then replaing m and m ′ b y the in tersetion p oin t u giv es a new quadruple with the same t . Hene, w e ma y assume that a and m are on the same side of the geo desi line L ′ through a ′ and m ′ . If [ a, m ] do es not en ter N ǫ − η C in the sphere ∂ B ( a ′ , ǫ − η ) , then dene a ∗ = a . Otherwise, replae a b y the p oin t a ∗ at distane equal to d ( a, a ′ ) from a ′ , su h that the geo desi segmen t b et w een a ∗ and m go es through the p oin t z ∗ ∈ ∂ B ( a ′ , ǫ − η ) ∩ ∂ N ǫ − η L ′ (on the same side of L ′ as a ). This giv es a new quadruple ( a ∗ , a ′ , m, m ′ ) satisfying the same prop erties, whose t has not dereased, b y on v exit y . Replae a ′ b y a ′ ∗ and a ∗ b y a ∗∗ su h that ∠ a ′ ∗ ( a ∗∗ , m ′ ) = π 2 , d ( a ∗∗ , a ′ ∗ ) = d ( a, a ′ ) , and a ∗ ∈ [ a ∗∗ , a ′ ∗ ] . Clearly , this do es not derease t . No w replae a ∗∗ b y the p oin t a ∗∗∗ su h that d ( a ∗∗∗ , a ′ ∗ ) = ǫ and [ a ∗∗ , a ′ ∗ ] ⊂ [ a ∗∗∗ , a ′ ∗ ] . Let m ∗ b e the p oin t on N ǫ/ 2 C su h that there is a geo desi line through a ∗∗∗ and m ∗ whi h is tangen t to N ǫ/ 2 C at m ∗ . Let m ′ ∗ b e its losest p oin t in L ′ . Again, the v alue of t for the quadruple ( a ∗∗∗ , a ′ ∗ , m ∗ , m ′ ∗ ) has not dereased. Hene, after these redutions, w e ma y assume that X = H 2 R , that the quadrilateral ( a, a ′ , m, m ′ ) has righ t angles at a ′ , m ′ , m , and that d ( a, a ′ ) = 2 d ( m, m ′ ) = ǫ . No w, let ℓ = d ( m, a ) − t b e the distane b et w een m and the p oin t at distane ǫ − η from [ a ′ , m ′ ] . An easy omputation (see [Bea, page 157℄) sho ws that cosh( t + ℓ ) = sinh( ǫ ) sinh( ǫ/ 2) and cosh ℓ = sinh( ǫ − η ) sinh( ǫ/ 2) . Consider the map f ǫ : s 7→ arcosh sinh( ǫ + s ) sinh( ǫ/ 2) . This funtion is inreasing and ona v e on [ − ǫ/ 2 , 0] , with f ǫ ( − ǫ/ 2) = 0 . By ona vit y , the graph of f ǫ on [ − ǫ/ 2 , 0] is ab o v e the line passing through its endp oin ts ( − ǫ/ 2 , 0) and (0 , f ǫ (0)) . Hene, for ev ery s in [0 , ǫ/ 2] , w e ha v e f ǫ (0) − f ǫ ( − s ) ≤ c ′′ ( ǫ ) s . Therefore t = f ǫ (0) − f ǫ ( − η ) ≤ c ′′ ( ǫ ) η as η ≤ ǫ/ 2 . This pro v es our laim, and ends the pro of of Lemma 2.6.  Here is a ner v ersion of Lemma 2.5 whi h sho ws that the en try p oin ts of a geo desi whi h en ters an ǫ -on v ex set for a long enough time and that of an y nearb y geo desi are lose. F or ev ery ǫ > 0 , w e dene c ′ 2 ( ǫ ) = max n c ′′ ( ǫ ) + 1 , 2 c ′ 1 ( ǫ ) ǫ , r cosh ǫ cosh ǫ − 1 sinh c ′ 1 ( ǫ ) c ′ 1 ( ǫ ) o . (- 3 -) Lemma 2.7 F or every ǫ > 0 , every ξ 0 in X ∪ ∂ ∞ X , every  onvex subset C in X , and al l ge o desi r ays or lines γ , γ ′ in X whih start at ξ 0 and enter N ǫ C at the p oints x, x ′ in X r esp e tively, if the length of γ ′ ∩ N ǫ C is at le ast c 0 ( ǫ ) , then we have d ( x, x ′ ) ≤ c ′ 2 ( ǫ ) d ( x, γ ′ ) . 12 Remarks. (1) Without assuming that the geo desi ra y or line γ ′ has a suien tly big p enetration distane inside N ǫ C , the result is false. (2) The urv ature assumption is neessary , as an b e seen b y onsidering geo desis whi h en ter a half-plane in R 2 almost parallel to the b oundary . Pro of. Let ǫ > 0 and assume that ξ 0 , C, γ , γ ′ , x, x ′ are as in the statemen t. W e ma y assume that x 6 = x ′ . In partiular ξ 0 / ∈ C . Let p ′ b e the p oin t of γ ′ the losest to x . Let [ x ′ , y ′ ] b e the in tersetion of γ ′ with N ǫ C (or [ x ′ , y ′ [ with y ′ ∈ ∂ ∞ X if γ ′ ∩ N ǫ C is un b ounded). Case 1: Assume rst that p ′ do es not b elong to [ ξ 0 , x ′ ] . If d ( x ′ , p ′ ) ≤ ǫ 2 c ′′ ( ǫ ) , then let a 0 b e the p oin t p ′ . Otherwise let a 0 b e the p oin t in [ x ′ , y ′ [ at distane ǫ 2 c ′′ ( ǫ ) from x ′ . This p oin t exists and is at distane at least ǫ 2 c ′′ ( ǫ ) ≥ d ( a 0 , x ′ ) from y ′ , as d ( x ′ , y ′ ) ≥ c 0 ( ǫ ) ≥ ǫ c ′′ ( ǫ ) . By Lemma 2.6, w e ha v e d ( a 0 , C ) ≤ ǫ − 1 c ′′ ( ǫ ) d ( a 0 , x ′ ) . Hene, if a 0 = p ′ , then, as d ( x, C ) = ǫ , w e ha v e d ( x, p ′ ) ≥ 1 c ′′ ( ǫ ) d ( p ′ , x ′ ) . So that d ( x, x ′ ) ≤ d ( x, p ′ ) + d ( p ′ , x ′ ) ≤ (1 + c ′′ ( ǫ )) d ( x, p ′ ) , whi h pro v es the result, b y the denition of c ′ 2 ( ǫ ) . If a 0 6 = p ′ , then p ′ / ∈ [ a 0 , ξ 0 [ . Let us pro v e that d ( x, p ′ ) ≥ ǫ 2 . This implies, b y Lemma 2.5 , that d ( x, x ′ ) ≤ c ′ 1 ( ǫ ) ≤ 2 c ′ 1 ( ǫ ) ǫ d ( p ′ , x ) , whi h pro v es the result, b y the denition of c ′ 2 ( ǫ ) . Let b 0 b e the p oin t in [ x ′ , y ′ ] at distane ǫ 2 c ′′ ( ǫ ) from y ′ (or b 0 = y ′ if y ′ is at innit y). By Lemma 2.6, w e ha v e max { d ( a 0 , C ) , d ( b 0 , C ) } ≤ ǫ − 1 c ′′ ( ǫ ) min { d ( a 0 , x ′ ) , d ( b 0 , y ′ ) } = ǫ 2 . If p ′ ∈ [ a 0 , b 0 ] , then b y on v exit y d ( p ′ , C ) ≤ ǫ 2 . As d ( x, C ) = ǫ , this implies that d ( x, p ′ ) ≥ ǫ 2 , as w an ted. If otherwise p ′ / ∈ [ a 0 , b 0 ] , then assume b y absurd that d ( x, p ′ ) < ǫ 2 . Let z b e the p oin t in [ x, ξ 0 [ whose losest p oin t to [ p ′ , ξ 0 [ is b 0 . By on v exit y , d ( z , b 0 ) < ǫ 2 . Hene d ( z , C ) ≤ d ( z , b 0 ) + d ( b 0 , C ) < ǫ , whi h on tradits the fat that γ en ters N ǫ C at x . Case 2: Assume no w that p ′ b elongs to [ ξ 0 , x ′ ] . Let a ′ and b ′ b e the p oin ts of C the losest to x ′ and y ′ resp etiv ely . They are at distane ǫ > 0 from x ′ and y ′ resp etiv ely (exept that b ′ = y ′ if y ′ is at innit y). Let φ b e the omparison angle at x ′ b et w een the geo desi segmen ts [ x ′ , a ′ ] and [ x ′ , y ′ [ . W e laim that sin φ ≤ 1 √ cosh ǫ . T o pro v e this laim, if y ′ ∈ X , w e onstrut a omparison quadrilateral with v erties x ′ , a ′ , b ′ , y ′ ∈ H 2 R b y gluing together the omparison triangles ( x ′ , a ′ , y ′ ) of ( x ′ , a ′ , y ′ ) and ( a ′ , b ′ , y ′ ) of ( a ′ , b ′ , y ′ ) along their isometri edges [ a ′ , y ′ ] . If y ′ / ∈ X , then b ′ = y ′ , and the ab o v e quadrilateral is replaed b y the omparison triangle with v erties x ′ , a ′ , y ′ ∈ H 2 R ∪ {∞} . By omparison, all angles in the quadrilateral in H 2 R with v erties x ′ , a ′ , b ′ , y ′ are greater than or equal to those in the quadrilateral in X with v erties x ′ , a ′ , b ′ , y ′ . In partiular, if the angle at x ′ is φ , w e ha v e φ ≤ φ . If the quadrilateral with v erties x ′ , a ′ , b ′ , y ′ is replaed b y the one with v erties x ′ , a ′ ∗ , b ′ ∗ , y ′ with d ( x ′ , a ′ ∗ ) = ǫ = d ( y ′ , b ′ ∗ ) and righ t angles at a ′ ∗ and b ′ ∗ , the angle φ ∗ at x ′ of this quadrilateral is at least φ . F urthermore, this quadrilateral is symmetri: the angle at y ′ is also φ ∗ . Th us, w e get an upp er b ound for φ b y estimating φ ∗ . 13 Let [ m, m ′ ] b e the ommon p erp endiular segmen t b et w een [ x ′ , y ′ ] and [ a ′ ∗ , b ′ ∗ ] , with m ∈ [ x ′ , y ′ ] . W e ha v e (see for instane [Bea , page 157℄), sin φ ∗ = cosh d ( m, m ′ ) cosh ǫ and co sh d ( x ′ , m ) = sinh ǫ sinh d ( m, m ′ ) . Hene, as d ( x ′ , y ′ ) ≥ c 0 ( ǫ ) , sin φ ∗ = q 1 + (sinh 2 ǫ ) / (cosh 2 d ( x ′ , m )) cosh ǫ ≤ q 1 + (sinh 2 ǫ ) / (cosh 2 ( c 0 ( ǫ ) / 2)) cosh ǫ ≤ 1 √ cosh ǫ , as c 0 ( ǫ ) ≥ ǫc ′′ ( ǫ ) ≥ 2 arcosh( √ 2 cosh( ǫ/ 2)) . This pro v es the laim. By on v exit y , the omparison angle at x ′ b et w een the geo desi segmen ts [ x ′ , x ] and [ x ′ , a ′ ] is at most π 2 . Hene the omparison angle θ at x ′ b et w een [ x ′ , x ] and [ x ′ , ξ 0 [ is at least π − π 2 − φ = π 2 − φ . In partiular, 1 sin θ ≤ 1 sin( π 2 − φ ) = 1 p 1 − sin 2 φ ≤ r cosh ǫ cosh ǫ − 1 . With x, x ′ as ab o v e, onsider ξ 0 in H 2 R ∪ ∂ ∞ H 2 R su h that ∠ x ′ ( x, ξ 0 ) = θ and d ( x ′ , ξ 0 ) = d ( x ′ , ξ 0 ) . By omparison, the p oin t p ′ on [ x ′ , ξ 0 ] the losest to x is at distane from x at most equal to d ( x, p ′ ) . If p ′ = x ′ , then d ( x, p ′ ) ≥ d ( x, p ′ ) = d ( x, x ′ ) , whi h implies the result, as c ′ 2 ( ǫ ) ≥ 1 . Otherwise, the angle ∠ p ′ ( x, x ′ ) is at least π 2 (equalit y holds if p ′ 6 = ξ 0 ). By the form ulae in righ t-angled h yp erb oli triangles, w e ha v e sinh d ( x, p ′ ) ≥ sinh d ( x, x ′ ) sin θ . As losest p oin t maps do not inrease distanes, w e ha v e d ( x, p ′ ) ≤ d ( x, x ′ ) ≤ c ′ 1 ( ǫ ) . In partiular sinh d ( x, p ′ ) ≤ sinh c ′ 1 ( ǫ ) c ′ 1 ( ǫ ) d ( x, p ′ ) b y on v exit y of the map t 7→ sinh t on [0 , + ∞ [ . Hene d ( x, x ′ ) = d ( x, x ′ ) ≤ sinh d ( x, x ′ ) ≤ sinh d ( x, p ′ ) sin θ ≤ sinh d ( x, p ′ ) sin θ ≤ c ′ 2 ( ǫ ) d ( x, p ′ ) , b y the denition of c ′ 2 ( ǫ ) .  In general, there is no estimate analogous to Lemma 2.7 for the distane b et w een the p oin ts y , y ′ where t w o geo desi ra ys or lines γ , γ ′ starting from a p oin t ξ 0 exit an ǫ -on v ex subset N ǫ C . F or instane, the geo desi line γ ould b e tangen t to N ǫ C , and γ ′ ould en ter for a long time in N ǫ C , so that y and y ′ w ould not b e lose. But the result is not true ev en if w e assume that b oth γ and γ ′ meet N ǫ C in a long segmen t. Here is a oun terexample when X is a tree (but this phenomenon is not sp ei to trees). Let γ , γ ′ b e t w o geo desi lines in a tree X , oiniding on their negativ e subra ys, starting at ξ 0 ∈ ∂ ∞ X , and with disjoin t p ositiv e subra ys. Let ǫ = η = 1 , and C = γ ′ ([ − ℓ, + ℓ ]) . Then the en tering p oin ts of γ , γ ′ in N ǫ C are x = x ′ = γ ′ ( − ℓ − 1) . Besides, y = γ (1) , y ′ = γ ′ ( ℓ + 1) and d ( y , γ ′ ) ≤ 1 . But w e ha v e d ( y , y ′ ) = ℓ + 2 , whi h go es to + ∞ as ℓ → + ∞ . C γ ′ 0 y x = x ′ ξ 0 y ′ γ 14 This explains the di hotom y in the follo wing result on the exiting p oin ts from an ǫ - on v ex sets of t w o geo desi lines whi h start from the same p oin t at innit y . F or ev ery ǫ, η > 0 , w e dene h ′ ( ǫ, η ) = max  2 η + max { 0 , − 2 log ǫ 2 } , η + c ′ 1 ( ǫ ) + c 0 ( ǫ )  (- 4 -) and c ′ 3 ( ǫ ) = 3 + 2 c ′ 1 ( ǫ ) ǫ . (- 5 -) Lemma 2.8 L et ǫ, η > 0 . L et C b e a  onvex subset in X , ξ 0 ∈ X ∪ ∂ ∞ X , and γ , γ ′ ge o desi r ays or lines starting fr om ξ 0 . If γ enters N ǫ C at a p oint x ∈ X and exits N ǫ C at a p oint y ∈ X suh that d ( x, y ) ≥ h ′ ( ǫ, η ) and d ( y , γ ′ ) ≤ η , then γ ′ me ets N ǫ C , entering it at a p oint x ′ ∈ X , exiting it at a p oint y ′ ∈ X ∪ ∂ ∞ X suh that d ( y , y ′ ) ≤ c ′ 3 ( ǫ ) d ( y , γ ′ ) or d ( x ′ , y ′ ) > d ( x, y ) . Pro of. Let p ′ b e the losest p oin t on γ ′ to y . Let q b e the losest p oin t on γ to p ′ . The p oin t q b elongs to [ y , ξ 0 ] and satises d ( y , q ) ≤ d ( y , p ′ ) ≤ η , as losest p oin t maps do not inrease the distanes. By the prop erties of geo desi triangles in CA T ( − 1) spaes, w e ha v e d ( p ′ , q ) ≤ arsinh 1 = log (1 + √ 2) . Let us rst pro v e that γ ′ meets N ǫ C . Let m b e the midp oin t of [ x, y ] . As d ( y , m ) = d ( x, y ) / 2 ≥ h ′ ( ǫ, η ) / 2 ≥ η ≥ d ( y , q ) , the p oin t q b elongs to [ m, y ] . F urthermore, d ( q , m ) = d ( y , m ) − d ( y , q ) ≥ h ′ ( ǫ, η ) / 2 − η ≥ − log ǫ 2 , b y the denition of h ′ ( ǫ, η ) . By Lemma 2.1, w e ha v e d ( m, γ ′ ) ≤ e − d ( q, m ) sinh d ( q , p ′ ) ≤ ǫ 2 . By Lemma 2.2 , as d ( x, y ) ≥ h ′ ( ǫ, η ) ≥ c 0 ( ǫ ) b y the denition of h ′ ( ǫ, η ) , w e ha v e d ( m, C ) ≤ ǫ 2 . Hene the p oin t m ′ of γ ′ the losest to m b elongs to N ǫ C , whi h is what w e w an ted. Let x ′ and y ′ b e the en tering p oin t in N ǫ C and exiting p oin t out of N ǫ C of γ ′ resp e- tiv ely , where y ′ ould for the momen t b e at innit y , in whi h ase the seond p ossibilit y b elo w w ould hold. ξ 0 γ γ ′ Case 2 x y γ γ ′ Case 1 ξ 0 x ′ p ′ x q y x ′ p ′ y ′ y ′ N ǫ C 15 Case 1 : Assume that p ′ / ∈ [ y ′ , ξ 0 ] . Let η ǫ = ǫ c ′′ ( ǫ ) / 2 . There are t w o sub ases. First assume that d ( y , p ′ ) ≥ ǫ/ 2 . Let t ǫ = max  η ǫ , − log ǫ 2  . Note that h ′ ( ǫ, η ) ≥ η ǫ + t ǫ + η b y the denition of h ′ ( ǫ, η ) , as c 0 ( ǫ ) ≥ ǫ c ′′ ( ǫ ) = 2 η ǫ . Hene w e ha v e d ( y , x ) − d ( y , q ) − t ǫ ≥ h ′ ( ǫ, η ) − η − t ǫ ≥ η ǫ ≥ 0 . Therefore, the p oin t y 0 in [ x, q ] at distane t ǫ of q exists and satises d ( y 0 , x ) ≥ η ǫ . F ur- thermore, d ( y 0 , q ) = t ǫ ≥ η ǫ and d ( x, q ) = d ( x, y ) − d ( y , q ) ≥ h ′ ( ǫ, η ) − η ≥ c 0 ( ǫ ) ≥ 2 η ǫ , b y the denition of h ′ ( ǫ, η ) . Let a 0 and b 0 b e the p oin ts in [ x, q ] at distane η ǫ from x and q resp etiv ely , whi h are at distane at least η ǫ from q and x resp etiv ely . By Lemma 2.6, w e ha v e d ( a 0 , C ) ≤ ǫ − η ǫ /c ′′ ( ǫ ) = ǫ/ 2 , and similarly d ( b 0 , C ) ≤ ǫ/ 2 . Note that y 0 b elongs to [ a 0 , b 0 ] . Hene b y on v exit y , w e ha v e d ( y 0 , C ) ≤ ǫ/ 2 . By Lemma 2.1, w e ha v e d ( y 0 , γ ′ ) ≤ e − t ǫ sinh d ( q , p ′ ) ≤ ǫ 2 . Therefore the p oin t q ′ on γ ′ the losest to y 0 b elongs to N ǫ C . As y ′ is the exiting p oin t of γ ′ from N ǫ C , it b elongs to [ q ′ , p ′ ] . As losest p oin t maps do not inrease the distanes, w e ha v e d ( p ′ , q ′ ) ≤ d ( y, y 0 ) . Hene d ( y , y ′ ) ≤ d ( y , p ′ ) + d ( p ′ , y ′ ) ≤ d ( y , p ′ ) + d ( p ′ , q ′ ) ≤ d ( y , p ′ ) + d ( y , y 0 ) ≤ d ( y , p ′ ) + d ( y, q ) + d ( q , y 0 ) ≤ 2 d ( y , p ′ ) + t ǫ ≤ (2 + 2 t ǫ /ǫ ) d ( y , p ′ ) ≤ c ′ 3 ( ǫ ) d ( y , p ′ ) , as it an b e  he k ed that 2 c ′ 1 ( ǫ ) /ǫ + 1 ≥ 2 t ǫ /ǫ . Assume no w that d ( y , p ′ ) ≤ ǫ/ 2 . Sine d ( x, y ) ≥ h ′ ( ǫ, η ) ≥ c 0 ( ǫ ) ≥ 2 η ǫ ≥ 2 c ′′ ( ǫ ) d ( y , p ′ ) , the p oin t y 0 in [ x, y ] at distane c ′′ ( ǫ ) d ( y , p ′ ) from y exists and d ( y 0 , x ) ≥ c ′′ ( ǫ ) d ( y , p ′ ) . Hene b y Lemma 2.6 , w e ha v e d ( y 0 , C ) ≤ ǫ − d ( y , p ′ ) . Let q ′ b e the p oin t on γ ′ the losest to y 0 . By on v exit y , q ′ is at distane at most d ( y , p ′ ) from y 0 , hene b elongs to N ǫ C . As y ′ is the exiting p oin t of γ ′ from N ǫ C , it b elongs to [ q ′ , p ′ ] . As losest p oin t maps do not inrease distanes, w e ha v e d ( q ′ , p ′ ) ≤ d ( y 0 , y ) . Hene, as ab o v e, d ( y , y ′ ) ≤ d ( y, p ′ ) + d ( y 0 , y ) ≤  1 + c ′′ ( ǫ )  d ( y , p ′ ) , whi h pro v es the result, b y the denition of c ′ 3 ( ǫ ) , as 2 c ′ 1 ( ǫ ) /ǫ + 1 ≥ c ′′ ( ǫ ) . Case 2 : Assume that p ′ ∈ ] y ′ , ξ 0 ] . Lemma 2.5 implies that d ( x, x ′ ) ≤ c ′ 1 ( ǫ ) . Note that p ′ / ∈ [ x ′ , ξ 0 ] . Otherwise, with q and s the losest p oin ts to p ′ and x ′ on γ resp etiv ely , w e w ould ha v e s / ∈ ] q , ξ 0 ] b y on v exit y . As q ∈ [ y , ξ 0 ] , w e w ould then ha v e d ( x, y ) ≤ d ( x, s ) + d ( q , y ) ≤ d ( x, x ′ ) + d ( p ′ , y ) ≤ c ′ 1 ( ǫ ) + η < h ′ ( ǫ, η ) , 16 b y the denition of h ′ ( ǫ, η ) , a on tradition. Assume rst that d ( y , p ′ ) < ǫ/ 2 . W e start b y observing that d ( p ′ , y ′ ) ≤ c ′′ ( ǫ ) d ( y , p ′ ) . Indeed, supp ose b y absurd that d ( p ′ , y ′ ) > c ′′ ( ǫ ) d ( y , p ′ ) . By on tin uit y of the losest p oin t maps, let y 0 b e a p oin t on γ that do es not b elong to N ǫ C , but is lose enough to y , so that the losest p oin t q ′ to y 0 on γ ′ b elongs to [ p ′ , y ′ ] and satises d ( y 0 , q ′ ) ≤ ǫ/ 2 and d ( q ′ , y ′ ) ≥ c ′′ ( ǫ ) d ( q ′ , y 0 ) . Hene, using the denition of h ′ ( ǫ, η ) , w e ha v e d ( y ′ , x ′ ) ≥ d ( q ′ , x ′ ) ≥ d ( p ′ , x ′ ) ≥ d ( x, y ) − d ( p ′ , y ) − d ( x, x ′ ) ≥ h ′ ( ǫ, η ) − η − c ′ 1 ( ǫ ) ≥ c 0 ( ǫ ) ≥ ǫ c ′′ ( ǫ ) ≥ 2 c ′′ ( ǫ ) d ( q ′ , y 0 ) . (- 6 -) Let a 0 and b 0 b e the p oin ts in [ x ′ , y ′ ] at distane c ′′ ( ǫ ) d ( q ′ , y 0 ) ≤ ǫ c ′′ ( ǫ ) / 2 from x ′ and y ′ resp etiv ely . The estimate (- 6 - ) implies that a 0 and b 0 are at distane at least c ′′ ( ǫ ) d ( q ′ , y 0 ) from y ′ and x ′ resp etiv ely . By Lemma 2.6 , w e ha v e d ( a 0 , C ) ≤ ǫ − d ( q ′ , y 0 ) and d ( b 0 , C ) ≤ ǫ − d ( q ′ , y 0 ) . Hene, the p oin t q ′ , whi h b elongs to [ a 0 , b 0 ] b y F orm ula (- 6 -) and the onstrution of q ′ , is b y on v exit y at distane at most ǫ − d ( q ′ , y 0 ) from C . Therefore b y the triangular inequalit y , d ( y 0 , C ) ≤ ǫ , whi h is a on tradition. Hene d ( p ′ , y ′ ) ≤ c ′′ ( ǫ ) d ( y , p ′ ) , and d ( y , y ′ ) ≤ d ( y , p ′ ) + d ( p ′ , y ′ ) ≤ (1 + c ′′ ( ǫ )) d ( y , p ′ ) , whi h pro v es the result, as in Case 1. Assume no w that d ( y , p ′ ) ≥ ǫ/ 2 . Supp ose rst that d ( p ′ , y ′ ) > d ( y , p ′ ) + c ′ 1 ( ǫ ) . Then, as p ′ ∈ [ x ′ , y ′ ] , d ( x ′ , y ′ ) = d ( x ′ , p ′ ) + d ( p ′ , y ′ ) ≥ d ( p ′ , y ′ ) + d ( x, y ) − d ( y , p ′ ) − d ( x, x ′ ) > d ( x, y ) , whi h is one of the t w o p ossible onlusions. Otherwise, d ( y , y ′ ) ≤ d ( y, p ′ ) + d ( p ′ , y ′ ) ≤ 2 d ( y, p ′ ) + c ′ 1 ( ǫ ) ≤ c ′ 3 ( ǫ ) d ( y , p ′ ) , b y the denition of c ′ 3 ( ǫ ) . This is the other p ossible onlusion.  Remark 2.9 The asymptoti b eha viour of the onstan ts when ǫ is v ery big or v ery small is as follo ws. • c 0 ( ǫ ) ∼ 3 ǫ as ǫ → + ∞ and lim ǫ → 0 c 0 ( ǫ ) = 4 log 2 ≈ 2 . 77 . • lim ǫ → + ∞ c ′ 1 ( ǫ ) = c ′ 1 ( ∞ ) = 2 log (1 + √ 2) ≈ 1 . 76 , and c ′ 1 ( ǫ ) ∼ − 2 log ǫ as ǫ → 0 . Note that ǫ 7→ c ′ 1 ( ǫ ) is dereasing. • lim ǫ → + ∞ c ′′ ( ǫ ) = 1 , and c ′′ ( ǫ ) ∼ 2 ǫ log(2 + √ 3) as ǫ → 0 . • F or ǫ big, c ′ 2 ( ǫ ) = c ′′ ( ǫ ) + 1 , hene lim ǫ → + ∞ c ′ 2 ( ǫ ) = 2 . F or ǫ > 0 small, c ′ 2 ( ǫ ) = r cosh ǫ cosh ǫ − 1 sinh c ′ 1 ( ǫ ) c ′ 1 ( ǫ ) ∼ √ 2 4 ǫ 3 log(1 /ǫ ) . • lim ǫ → + ∞ c ′ 3 ( ǫ ) = 3 , and c ′ 3 ( ǫ ) ∼ − 4 ǫ log ǫ as ǫ → 0 . • h ′ ( ǫ, η ) ∼ 3 ǫ as ǫ → + ∞ , and h ′ ( ǫ, η ) ∼ − 2 log ǫ as ǫ → 0 , uniformly on ompat subsets of η 's. 17 When ǫ go es to + ∞ , c ′ 1 ( ǫ ) and c ′ 3 ( ǫ ) ha v e nite limits, and the limiting v alues apply for the horoball ase, see Lemmas 2.11 and 2.14 b elo w. On the other hand, the onstan ts c 0 ( ǫ ) and h ′ ( ǫ, η ) b eha v e badly as ǫ → ∞ , and w e will impro v e them in Setion 2.3 . When X is a tree, the onstan ts c ′ 3 ( ǫ ) and h ′ ( ǫ, η ) an b e simplied, w e an tak e c ′ 3 ( ǫ ) = 2 and an y h ′ ( ǫ, η ) > 2 η , as the follo wing more preise result sho ws, impro ving Lemma 2.8 for trees. Note that the v ersions of Lemmas 2.5 and 2.7 for trees simply sa y that w e an tak e c 0 ( ǫ ) = ǫ , and c ′ 1 ( ǫ ) = c ′ 2 ( ǫ ) = 0 , sine for ev ery p oin t or end ξ 0 of a (real) tree, for ev ery on v ex subset C , for all geo desi ra ys or lines γ , γ ′′ starting from ξ 0 and en tering C in x, x ′ resp etiv ely , w e ha v e x = x ′ . Remark 2.10 L et X b e an R -tr e e and ǫ > 0 . L et C b e a  onvex subset in X , ξ 0 ∈ X ∪ ∂ ∞ X , and γ , γ ′ ge o desi r ays or lines starting fr om ξ 0 . If γ enters N ǫ C at a p oint x ∈ X and exits N ǫ C at a p oint y ∈ X suh that d ( x, y ) > 2 d ( y , γ ′ ) , then γ ′ me ets N ǫ C , entering it at x ′ = x , exiting it at a p oint y ′ (p ossibly at innity) suh that d ( y , y ′ ) ≤ 2 d ( y , γ ′ ) or d ( x ′ , y ′ ) > d ( x, y ) . Pro of. Let p ′ b e the losest p oin t to y on γ ′ . Note that p ′ b elongs to ] ξ 0 , y ] , as X is a tree and γ ′ also starts from ξ 0 . If p ′ ∈ ] ξ 0 , x [ , then d ( y , γ ′ ) > d ( x, y ) , a on tradition. Hene p ′ ∈ [ x, y ] ⊂ N ǫ C , and γ ′ en ters N ǫ C at x ′ = x . Supp ose rst that d ( x, y ) < 2 ǫ . Then the losest p oin t z to y in C do es not b elong to [ x, y ] . Let q b e the midp oin t of [ x, y ] , whi h is also the losest p oin t to z on [ x, y ] . As d ( x, y ) > 2 d ( y , γ ′ ) , the p oin t p ′ b elongs to ] q , y ] , hene d ( y , y ′ ) = 2 d ( y, γ ′ ) , whi h is ne. Assume no w that d ( x, y ) ≥ 2 ǫ . If x ǫ and y ǫ are the p oin ts in [ x, y ] at distane ǫ from x and y resp etiv ely , then [ x, y ] ∩ C = [ x ǫ , y ǫ ] . If p ′ b elongs to ] y ǫ , y ] , then y ǫ is also the losest p oin t to y ′ in C , and d ( p ′ , y ) = d ( p ′ , y ′ ) , so that d ( y , y ′ ) = 2 d ( y , γ ′ ) , whi h is ne. Otherwise, w e ha v e d ( y , γ ′ ) ≥ ǫ . If d ( x ′ , y ′ ) ≤ d ( x, y ) , then d ( p ′ , y ′ ) ≤ d ( p ′ , y ) . Hene d ( y , y ′ ) = d ( y , p ′ ) + d ( p ′ , y ′ ) ≤ 2 d ( y, p ′ ) = 2 d ( y, γ ′ ) .  2.3 Hitting horoballs As sho wn in Remark 2.9, the onstan ts c 0 ( ǫ ) and h ′ ( ǫ, η ) , used to desrib e the p enetration of geo desi lines inside ǫ -on v ex subsets, do not ha v e a nite limit as ǫ go es to + ∞ . Horoballs are ǫ -on v ex subsets for ev ery ǫ , and w e ould use for instane ǫ = 1 in these onstan ts to get n umerial v alues. But in order to get b etter v alues, w e will pro v e analogs for horoballs of the lemmas 2.5, 2.6, 2.7 and 2.8. The pro ofs of the lemmas b elo w follo w the same lines as the ones for the general ase of ǫ -on v ex subsets giv en in Setion 2.2, with man y simpliations. As c ′ 1 ( ǫ ) tends to c ′ 1 ( ∞ ) = 2 log(1 + √ 2) , the next lemma follo ws b y passing to the limit in Lemma 2.5 . It is not hard to see (for instane b y onsidering the real h yp erb oli plane) that the onstan t c ′ 1 ( ∞ ) is optimal. Lemma 2.11 F or every hor ob al l H in X , for every ξ 0 in ( X ∪ ∂ ∞ X ) − ( H ∪ H [ ∞ ]) , for al l ge o desi r ays or lines γ and γ ′ starting fr om ξ 0 and entering H in x and x ′ r esp e tively, we have d ( x, x ′ ) ≤ c ′ 1 ( ∞ ) = 2 log(1 + √ 2) .  18 The follo wing result, Lemma 2.12 , impro v es Lemma 2.6 for horoballs, and sa ys that when the ǫ -on v ex subset under onsideration is a horoball, w e an replae c 0 ( ǫ ) b y c 0 ( ∞ ) = 4 . 056 , (- 7 -) and c ′′ ( ǫ ) b y c ′′ ( ∞ ) = 3 2 . Lemma 2.13 b elo w is the analog of Lemma 2.7 for horoballs, and sa ys that when the ǫ -on v ex subset under onsideration is a horoball, w e an replae c 0 ( ǫ ) b y c 0 ( ∞ ) = 4 . 056 and c ′ 2 ( ǫ ) b y c ′ 2 ( ∞ ) = 5 2 . (- 8 -) Note that c ′′ ( ∞ ) , c 0 ( ∞ ) and c ′ 2 ( ∞ ) are not limits as ǫ go es to ∞ of c 0 ( ǫ ) and c ′ 2 ( ǫ ) , but this notation will b e useful in Setion 4. Lemma 2.12 F or every hor ob al l H , for every a and b in ∂ H with d ( a, b ) ≥ c 0 ( ∞ ) , for every a 0 in [ a, b ] , we have a 0 ∈ H [ 2 3 min { d ( a 0 , a ) , d ( a 0 , b ) } ] . Pro of. Let ξ = H [ ∞ ] b e the p oin t at innit y of H . By symmetry , w e ma y assume that ℓ = d ( a 0 , a ) = min { d ( a 0 , a ) , d ( a 0 , b ) } . Let ( a, b, ξ = ∞ ) b e a omparison triangle of ( a, b, ξ ) in H 2 R . By omparison, the dierene ℓ ′ of the heigh ts of a 0 and a with resp et to ξ is bigger than the orresp onding quan tit y ℓ ′ for the omparison p oin ts a 0 and a . Th us, in order to sho w that ℓ ′ ≥ 2 3 ℓ , it sues to sho w that ℓ ′ ≥ 2 3 ℓ , and the question redues to the ase X = H 2 R . W e assume that [ b, a ] lies on the unit irle, with a (hene a 0 , as a and b ha v e the same (Eulidean) v ertial o ordinate) in the losed p ositiv e quadran t. Let s b e the (Eulidean) v ertial o ordinate of a 0 and t the one of a , with 0 < t ≤ s ≤ 1 . An easy ompu- tation in h yp erb oli geometry (see also the pro of of Lemma 2.1) giv es ℓ ′ = log s t and ℓ = arsinh √ 1 − t 2 t − arsinh √ 1 − s 2 s = log s t + log 1 + √ 1 − t 2 1 + √ 1 − s 2 . a ℓ ′ ℓ 1 ξ t s a 0 Hene, to pro v e that ℓ ≤ 3 2 ℓ ′ , w e only ha v e to sho w that log 1+ √ 1 − t 2 1+ √ 1 − s 2 ≤ 1 2 log s t , whi h is equiv alen t to √ t (1 + √ 1 − t 2 ) ≤ √ s (1 + √ 1 − s 2 ) . The map f : x 7→ √ x (1 + √ 1 − x 2 ) on [0 , 1] is inreasing from f (0) = 0 to f ( √ 5 3 ) , and then dereasing to f (1) = 1 . Let t ′ = 0 . 258 73 . As f ( t ′ ) < 1 and s ≥ t , to pro v e that f ( t ) ≤ f ( s ) , it is suien t to sho w that t ≤ t ′ . Let a ′ and b ′ b e the t w o p oin ts of the unit irle at (Eulidean) heigh t t ′ . As a and b are at the same (Eulidean) heigh t t on the unit irle, to pro v e that t ≤ t ′ , w e only ha v e to sho w that d ( a ′ , b ′ ) ≤ d ( a, b ) . By the denition of c 0 ( ∞ ) , w e ha v e d ( a ′ , b ′ ) = 2 arsinh p 1 − t ′ 2 t ′ ≤ c 0 ( ∞ ) ≤ d ( a, b ) . Hene the result follo ws.  19 Lemma 2.13 F or every hor ob al l H in X , for every ξ 0 in X ∪ ∂ ∞ X , for al l ge o desi r ays or lines γ and γ ′ starting fr om ξ 0 and entering H in x ∈ X and x ′ ∈ X r esp e tively, if the length of γ ′ ∩ H is at le ast c 0 ( ∞ ) , then d ( x, x ′ ) ≤ 5 2 d ( x, γ ′ ) . Pro of. Let p ′ b e the p oin t of γ ′ the losest to x . Let ξ b e the p oin t at innit y of H . Dene y ′ b y [ x ′ , y ′ ] = γ ′ ∩ H if this in tersetion is b ounded, and y ′ = ξ otherwise. W e ma y assume that x 6 = x ′ . In partiular, ξ 0 / ∈ H ∪ { ξ } . Assume rst that p ′ do es not b elong to [ ξ 0 , x ′ [ . As losest p oin t pro jetions do not inrease distanes and b y Lemma 2.11 , w e ha v e d ( x ′ , p ′ ) ≤ c ′ 1 ( ∞ ) , and sine d ( x ′ , y ′ ) ≥ c 0 ( ∞ ) ≥ 2 c ′ 1 ( ∞ ) , the p oin t p ′ b elongs to H , and d ( p ′ , y ′ ) ≥ d ( p ′ , x ′ ) . Let z b e the p oin t of in tersetion of ] ξ , x ′ ] with the horosphere en tered at ξ passing through p ′ , so that in partiular d ( x, p ′ ) ≥ d ( x ′ , z ) . By Lemma 2.12 , w e ha v e d ( x ′ , z ) ≥ 2 3 d ( x ′ , p ′ ) . Hene d ( x, x ′ ) ≤ d ( x, p ′ ) + d ( p ′ , x ′ ) ≤ d ( x, p ′ ) + 3 2 d ( x ′ , z ) ≤ 5 2 d ( x, p ′ ) . Assume no w that p ′ b elongs to [ ξ 0 , x ′ [ . Let β b e the omparison angle at x ′ b et w een the (non trivial) geo desi segmen ts or ra ys [ x ′ , y ′ [ and [ x ′ , ξ [ . By omparison, β is at most the angle β b et w een [ x ′ , y ′ [ and [ x ′ , ξ [ , where x ′ and y ′ are t w o p oin ts, at distane d ( x ′ , y ′ ) , on an horosphere in H 2 R en tered at ξ . An easy omputation in the upp er half spae mo del sho ws that tan β =  sinh 1 2 d ( x ′ , y ′ )  − 1 ≤  sinh  2 log(1 + √ 2)   − 1 ≤ 1 √ 3 . As 0 ≤ β ≤ π 2 , this implies that β ≤ β ≤ π 6 . Let α b e the omparison angle at x ′ b et w een the (non trivial) geo desi segmen ts [ x ′ , p ′ ] and [ x ′ , x ] , whi h is at most π 2 , as p ′ is the losest p oin t to x on ] x ′ , ξ 0 ] . As the geo desi segmen t [ x, x ′ ] lies in H , w e ha v e α ≥ π − π 2 − β ≥ π 3 . By Lemma 2.11 , w e ha v e d ( x, p ′ ) ≤ d ( x, x ′ ) ≤ c ′ 1 ( ∞ ) . Using the form ulae for righ t-angled h yp erb oli triangles (see [Bea℄) and the omparison triangle in H 2 R to the triangle ( x, x ′ , p ′ ) in X , w e ha v e, b y on v exit y of t 7→ sinh t , d ( x, x ′ ) ≤ sinh d ( x, x ′ ) ≤ 1 sin α sinh d ( x, p ′ ) ≤ 2 √ 3 sinh c ′ 1 ( ∞ ) c ′ 1 ( ∞ ) d ( x, p ′ ) ≤ 5 2 d ( x, p ′ ) . This pro v es the result.  The follo wing Lemma is the analog of Lemma 2.8 for horoballs. It sa ys that when the ǫ -on v ex subset under onsideration is a horoball, w e an replae c ′ 3 ( ǫ ) and h ′ ( ǫ, η ) b y c ′ 3 ( ∞ ) = 5 2 and h ′ ( ∞ , η ) = 3 η + c 0 ( ∞ ) + c ′ 1 ( ∞ ) ≈ 3 η + 5 . 8 188 , (- 9 -) and that the rst of the t w o p ossible onlusions of Lemma 2.8 alw a ys holds. Note that c ′ 3 ( ∞ ) is not the limit as ǫ go es to + ∞ of c ′ 3 ( ǫ ) , and that h ′ ( ǫ, η ) div erges as ǫ → ∞ . Ho w ev er, in b oth ases, this notation will b e useful in Setion 4. 20 Lemma 2.14 F or every hor ob al l H in X , for every ξ 0 in X ∪ ∂ ∞ X , for al l ge o desi r ays or lines γ , γ ′ starting fr om ξ 0 , if γ enters H at a p oint x ∈ X and exits H at a p oint y ∈ X , and if d ( x, y ) ≥ h ′ ( ∞ , d ( y , γ ′ )) , then γ ′ me ets H , exiting it at a p oint y ′ ∈ X suh that d ( y , y ′ ) ≤ 5 2 d ( y , γ ′ ) . Pro of. Let ξ b e the p oin t at innit y of H , let p b e the losest p oin t on [ x, y ] from ξ , and let p x and p y b e the in tersetion of the horosphere ∂ H p en tered at ξ passing through p with the geo desi ra ys [ x, ξ [ and [ y , ξ [ resp etiv ely . By omparison, w e ha v e d ( p x , p y ) ≤ 2 log (1 + √ 2) = c ′ 1 ( ∞ ) . Th us, the triangle inequalit y , along with the fat that p y is the losest p oin t to y on ∂ H p and the assumption on d ( x, y ) , giv es 2 min { d ( y , p ) , d ( x, p ) } ≥ d ( y , p y ) + d ( x, p x ) ≥ d ( x, y ) − 2 log(1 + √ 2) ≥ c 0 ( ∞ ) ≥ 3 . ∂ H ℓ H ( γ ) γ y p x ξ x p y p ph H ( γ ) 2 In partiular, as d ( x, y ) ≥ c 0 ( ∞ ) , Lemma 2.12 implies that p b elongs to H [1] . By Lemma 2.1 and the assumption on d ( x, y ) , w e ha v e d ( p, γ ′ ) ≤ 1 2 e − d ( y,p )+ d ( y,γ ′ ) ≤ 1 2 e − 1 2 d ( x,y )+l og(1+ √ 2)+ d ( y,γ ′ ) ≤ 1 2 . This implies that γ ′ meets H , b eause N 1 2 ( H [1]) is on tained in H . Let x ′ and y ′ b e the en tering p oin t in H and the exiting p oin t out of H of γ ′ , resp etiv ely . Let p ′ b e the p oin t on γ ′ the losest to y . Case 1 : Assume that p ′ / ∈ [ y ′ , ξ 0 ] . Note that d ( x, y ) − 3 2 d ( y , p ′ ) ≥ 3 2 d ( y , p ′ ) ≥ 0 , as d ( x, y ) ≥ 3 d ( y , γ ′ ) . Hene, there is a p oin t y 0 in [ x, y ] at distane 3 2 d ( y , p ′ ) of y whi h satises d ( x, y 0 ) ≥ 3 2 d ( y , p ′ ) . By Lemma 2.12 , w e ha v e y 0 ∈ H [ d ( y , p ′ )] . Let q ′ b e the p oin t of γ ′ the losest to y 0 . By on v exit y , w e ha v e d ( y 0 , q ′ ) ≤ d ( y , p ′ ) . Hene q ′ b elongs to H . By the in termediate v alue theorem, the p oin t y ′ b elongs to [ q ′ , p ′ ] . As losest p oin t maps do not inrease the distanes, w e ha v e d ( p ′ , q ′ ) ≤ d ( y, y 0 ) = 3 2 d ( y , p ′ ) . Therefore, d ( y , y ′ ) ≤ d ( y , p ′ ) + d ( p ′ , y ′ ) ≤ d ( y , p ′ ) + d ( p ′ , q ′ ) ≤ 5 2 d ( y , p ′ ) , whi h pro v es the result. Case 2 : Assume that p ′ ∈ ] y ′ , ξ 0 ] . By the same argumen t as in Case 2 of the pro of of Lemma 2.8, w e ha v e p ′ / ∈ [ x ′ , ξ 0 ] . If d ( p ′ , y ′ ) ≤ 3 2 d ( y , p ′ ) , then d ( y , y ′ ) ≤ 5 2 d ( y , p ′ ) , and the result is pro v ed. Therefore, assume b y absurd that d ( p ′ , y ′ ) > 3 2 d ( y , p ′ ) . By the on tin uit y of the losest p oin t maps, there exists a p oin t y 0 in γ that do es not b elong to H , whose losest p oin t q ′ on γ ′ , whi h lies in γ ′ − ] p ′ , ξ 0 ] , satises d ( q ′ , y ′ ) ≥ 3 2 d ( y 0 , q ′ ) and 21 d ( y 0 , q ′ ) ≤ d ( y , p ′ ) + 1 2 c 0 ( ∞ ) . Lemma 2.11 implies that d ( x, x ′ ) ≤ c ′ 1 ( ∞ ) . Th us, b y the assumption on d ( x, y ) , d ( x ′ , y ′ ) ≥ d ( x ′ , q ′ ) ≥ d ( p ′ , x ′ ) ≥ d ( x, y ) − d ( x, x ′ ) − d ( y, p ′ ) ≥ 3 d ( y , p ′ ) + c 0 ( ∞ ) + c ′ 1 ( ∞ ) − c ′ 1 ( ∞ ) − d ( y, p ′ ) ≥ 2 d ( y , p ′ ) + c 0 ( ∞ ) ≥ max { 2 d ( y 0 , q ′ ) , c 0 ( ∞ ) } . In partiular, d ( y 0 , q ′ ) ≤ 2 3 d ( q ′ , x ′ ) and w e already had d ( y 0 , q ′ ) ≤ 2 3 d ( q ′ , y ′ ) . Hene, b y Lemma 2.12 , w e ha v e q ′ ∈ H [ d ( y 0 , q ′ )] . This implies that y 0 b elongs to H , a on tradition.  3 Prop erties of p enetration in ǫ -on v ex sets 3.1 P enetration maps Let X b e a prop er geo desi CA T ( − 1) spae, and ξ 0 ∈ X ∪ ∂ ∞ X . W e are in terested in on trolling the p enetration of geo desi ra ys or lines starting from ξ 0 in ǫ -on v ex subsets of X . One w a y to measure this p enetration is the in tersetion length. If C is a losed on v ex subset in X su h that ξ 0 / ∈ C ∪ ∂ ∞ C , w e dene a map ℓ C : T 1 ξ 0 X → [0 , + ∞ ] , alled the p enetr ation length map , whi h asso iates to ev ery γ in T 1 ξ 0 X the length of the in tersetion γ ∩ C (whi h is onneted b y on v exit y). When w e study sp ei geometri situations, su h as olletions of horoballs and ǫ - neigh b ourho o ds of geo desis, there are further natural w a ys of measuring the p enetration. These will b e used in man y appliations in Setion 5 and in [HPP ℄. If C is an ǫ -on v ex subset of X su h that ξ 0 / ∈ C ∪ ∂ ∞ C , w e will require our p enetration maps f : T 1 ξ 0 X → [0 , + ∞ ] in C to ha v e one or t w o of the follo wing prop erties, the rst one dep ending on a onstan t κ ≥ 0 . The sup norm of a real v alued funtion f on T 1 ξ 0 X is denoted b y k f k ∞ . ( i ) (P enetration prop ert y) k f − ℓ C k ∞ ≤ κ . ( ii ) (Lips hitz prop ert y) F or ev ery γ , γ ′ in T 1 ξ 0 X whi h in terset C , if γ ∩ C = [ a, b ] and γ ′ ∩ C = [ a ′ , b ′ ] with a, b, a ′ , b ′ in X , then | f ( γ ) − f ( γ ′ ) | ≤ 2 max  d ( a, a ′ ) , d ( b, b ′ )  . If C is an ǫ -on v ex subset of X su h that ξ 0 / ∈ C ∪ ∂ ∞ C , and f : T 1 ξ 0 X → [0 , + ∞ [ is a map whi h satises ( i ) for some κ ≥ 0 , w e sa y that f is a κ - p enetr ation map in (the ǫ - onvex set) C . W e also sa y that ( C, f ) is an ( ǫ, κ ) - p enetr ation p air . In the ondition ( ii ) , w e ould ha v e replaed 2 b y some λ ≥ 2 , but if f also satises the prop ert y ( i ) , then only λ = 2 is really relev an t in the large sale. Note that if ( C, f ) is an ( ǫ ′ , κ ′ ) -p enetration pair, if ǫ ′ ≥ ǫ and κ ′ ≤ κ , then ( C, f ) is an ( ǫ, κ ) -p enetration pair. If C is ∞ -on v ex and ( C, f ) is an ( ǫ, κ ) -p enetration pair in ev ery ǫ > 0 then f is a κ -p enetr ation map in (the ∞ - onvex set) C . P enetration maps in general ǫ -on v ex subsets. If C is a losed on v ex subset of X , the map ℓ C is in general not on tin uous on T 1 ξ 0 X , as an b e seen b y taking C to b e a geo desi segmen t of p ositiv e length. The follo wing result sho ws that the situation is nier for ǫ -on v ex subsets. Note that the statemen t of Lemma 3.1 is not true in R n (whi h is not a CA T ( − 1) spae). 22 Lemma 3.1 L et ǫ > 0 and let C b e an ǫ - onvex subset of X suh that ξ 0 / ∈ C ∪ ∂ ∞ C . The map ℓ C : T 1 ξ 0 X → [0 , + ∞ ] is a  ontinuous 0 -p enetr ation map in C satisfying the Lipshitz pr op erty ( ii ) . Pro of. The Lips hitz prop ert y ( ii ) of the p enetration length map ℓ C follo ws from the triangular inequalit y . It remains to sho w the on tin uit y of the map. Cho ose a on v ex subset C ′ su h that C = N ǫ ( C ′ ) , and note that b y the denition of the top ology of X ∪ ∂ ∞ X , the subsets C and C ′ ha v e the same p oin ts at innit y . Let γ 0 ∈ T 1 ξ 0 X , and let us pro v e that ℓ C is on tin uous at γ 0 . Assume rst that γ 0 (+ ∞ ) is a p oin t at innit y of C . Then there exists a geo desi ra y on tained in C ′ ending at this p oin t at innit y . As geo desi ra ys on v erging to the same p oin t at innit y b eome exp onen tially lose, this implies that ℓ C ( γ 0 ) = ∞ . Let A > 0 . As γ 0 ∩ C is the losure of γ 0 ∩ ◦ C , let [ x, y ] b e a geo desi segmen t of length A + 2 on tained in γ 0 ∩ ◦ C . Let η ∈ ]0 , 1] b e su h that the balls B and B ′ of radius η and of en ter x and y resp etiv ely are on tained in ◦ C . If γ ∈ T 1 ξ 0 X is lose enough to γ 0 , then γ meets B and B ′ , and b y on v exit y , ℓ C ( γ ) ≥ A , whi h pro v es the result. Assume no w that γ 0 (+ ∞ ) is not a p oin t at innit y of C , but that γ 0 do es meet C . Then γ 0 ∩ C is a nonempt y ompat segmen t [ a, b ] . F or ev ery η > 0 , let a + , b + b e p oin ts in γ 0 − [ a, b ] , at distane at most η / 4 from a, b resp etiv ely , and, if d ( a, b ) > 0 , let a − , b − b e p oin ts in ] a, b [ at distane at most η / 4 from a, b resp etiv ely . As C is losed and γ 0 ∩ C is the losure of γ 0 ∩ ◦ C if a 6 = b , there exists η ′ ∈ ]0 , η / 4] su h that the balls B ( a + ) , B ( b + ) of radius η ′ and en ters a + , b + resp etiv ely are on tained in X − C and, if d ( a, b ) > 0 , the balls B ( a − ) , B ( b − ) of radius η ′ and en ters a − , b − resp etiv ely are on tained in the in terior of C . If γ ∈ T 1 ξ 0 X is lose enough to γ 0 , then γ meets B ( a + ) , B ( b + ) (and hene B ( a − ) , B ( b − ) b y on v exit y , if d ( a, b ) > 0 ). It is easy to see then that | ℓ C ( γ ) − ℓ C ( γ 0 ) | ≤ η . Assume no w that γ 0 do es not meet C . Let U, V b e neigh b orho o ds of the endp oin ts of γ 0 in X ∪ ∂ ∞ X that are disjoin t from C ∪ ∂ ∞ C . Let η > 0 b e su h that the η -neigh b orho o d of γ 0 is disjoin t from C , whi h exists, as inf x ∈ γ 0 d ( x, C ) > 0 . If γ ∈ T 1 ξ 0 X is lose enough to γ 0 , then (the image of ) γ lies in U ∪ V ∪ N η γ 0 , hene do es not meet C . So that ℓ C ( γ ) = ℓ C ( γ 0 ) = 0 .  In partiular, if H is a horoball su h that ξ 0 / ∈ H ∪ ∂ ∞ H , then ℓ H is a on tin uous 0 -p enetration map for H satisfying the Lips hitz prop ert y ( ii ) . Let C b e a on v ex subset of X su h that ξ 0 / ∈ C ∪ ∂ ∞ C . F or ev ery γ in T 1 ξ 0 X , let γ − = ξ 0 and γ + = γ (+ ∞ ) , and let q γ ± b e the losest p oin t on C to γ ± . Dene the b oundary-pr oje tion p enetr ation map bp C : T 1 ξ 0 X → [0 , + ∞ ] b y bp C ( γ ) = d ( q γ − , q γ + ) , with the ob vious on v en tion that bp C ( γ ) = + ∞ if q γ + is at innit y . Lemma 3.2 L et C b e an ǫ - onvex subset of X suh that ξ 0 / ∈ C ∪ ∂ ∞ C . The map bp C is a  ontinuous 2 c ′ 1 ( ǫ ) -p enetr ation map in C . Pro of. The on tin uit y of bp C follo ws from the on tin uit y of the pro jetion maps and the endp oin t maps. Let γ ∈ T 1 ξ 0 X , and let us pro v e that | bp C ( γ ) − ℓ C ( γ ) | ≤ 2 c ′ 1 ( ǫ ) . If γ + is a p oin t at innit y of C , then bp C ( γ ) = ℓ C ( γ ) = + ∞ , and the result is true. Otherwise, if γ 23 meets C , then γ en ters C at x and exits C at y , with x, y in X . By Lemma 2.5, w e hene ha v e | d ( x, y ) − d ( q γ − , q γ + ) | ≤ d ( x, q γ − ) + d ( q γ + , y ) ≤ 2 c ′ 1 ( ǫ ) , and the result follo ws. If γ do es not meet C , let [ p, q ] b e the shortest onneting segmen t b et w een a p oin t p in γ and a p oin t q in C . By angle omparison, the geo desi segmen t or ra y b et w een q and γ ± meets C exatly in q . Hene, b y Lemma 2.5, d ( q γ − , q γ + ) ≤ d ( q γ − , q ) + d ( q , q γ + ) ≤ 2 c ′ 1 ( ǫ ) . As ℓ C ( γ ) = 0 , the result follo ws.  P enetration maps in horoballs. If H is a horoball in X , with ξ its p oin t at innit y , su h that ξ 0 / ∈ H ∪ { ξ } , and if x 0 is an y p oin t in the b oundary of H in X , dene a 1 -Lips hitz map β H : X → [0 , + ∞ [ , alled the height map of H b y β H : x 7→ max { β ξ ( x 0 , x ) , 0 } , whose v alues are p ositiv e in the in terior of H , and 0 outside H . By on v en tion, dene β H ( ξ ) = + ∞ . F or ev ery γ in T 1 ξ 0 X , let p γ b e the losest p oin t to γ (+ ∞ ) on the geo desi line b et w een ξ 0 and ξ , with p γ = ξ if γ (+ ∞ ) = ξ . W e will study t w o p enetration maps asso iated with the heigh t map. The map ph H : T 1 ξ 0 X → [0 , + ∞ ] dened b y ph H ( γ ) = 2 su p t ∈ R β H ( γ ( t )) will b e alled the p enetr ation height map inside H . The map ipp H : T 1 ξ 0 X → [0 , + ∞ ] dened b y ipp H ( γ ) = 2 β H ( p γ ) will b e alled the inner-pr oje tion p enetr ation map inside H . Note that for ev ery t ≥ 0 and γ ∈ T 1 ξ 0 X , w e ha v e ph H [ t ] ( γ ) = m ax { 0 , ph H ( γ ) − 2 t } and ipp H [ t ] ( γ ) = m ax { 0 , ipp H ( γ ) − 2 t } . Lemma 3.3 L et H b e an hor ob al l in X , suh that ξ 0 / ∈ H ∪ ∂ ∞ H . The maps ph H , ipp H : T 1 ξ 0 X → [0 , + ∞ ] ar e  ontinuous 2 log(1 + √ 2) -p enetr ation maps for H , and ph H has the Lipshitz pr op erty ( ii ) . F urthermor e, k ph H − ipp H k ∞ ≤ 2 log (1 + √ 2) . Remark. In H n R , the equalit y ipp H ( γ ) = ph H ( γ ) + log 2 holds for an y horoball H with ξ 0 / ∈ H ∪ H [ ∞ ] and γ ∈ T 1 ξ 0 X meeting H . Th us, in H n R the map ipp H satises the Lips hitz prop ert y ( ii ) . W e do not kno w whether ( H , ipp H ) satises the Lips hitz prop ert y ( ii ) in general. Pro of. Let us pro v e that ( H , ph H ) satises the Lips hitz prop ert y ( ii ) . Let γ , γ ′ b e elemen ts in T 1 ξ 0 X su h that γ ∩ H = [ a, b ] and γ ′ ∩ H = [ a ′ , b ′ ] . Then, for ev ery x in [ a, b ] , if x ′ is the p oin t on [ a ′ , b ′ ] the losest to x , w e ha v e, with ξ the p oin t at innit y of H , | β ξ ( x, a ) − β ξ ( x ′ , a ) | = | β ξ ( x, x ′ ) | ≤ d ( x, x ′ ) ≤ max { d ( a, a ′ ) , d ( b, b ′ ) } b y on v exit y . T aking x the highest p oin t in [ a, b ] , and using a symmetry argumen t, the result follo ws. 24 Let us pro v e that ( H , ph H ) satises the P enetration prop ert y ( i ) with κ = c ′ 1 ( ∞ ) = 2 log(1 + √ 2) . Let γ ∈ T 1 ξ 0 X . Note that γ en ters the in terior of H if and only if ℓ H ( γ ) > 0 , and if and only if ph H ( γ ) > 0 . Hene w e ma y assume that γ meets H in a segmen t [ x, y ] . By the rst paragraph of the pro of of Lemma 2.14 , w e ha v e ph H ( γ ) ≤ ℓ H ( γ ) ≤ ph H ( γ ) + 2 log (1 + √ 2) . Let us pro v e that ( H , ipp H ) satises the P enetration prop ert y ( i ) with κ = 2 log(1 + √ 2) . Let γ ∈ T 1 ξ 0 X . If p γ = ξ , then ipp H ( γ ) = ℓ H ( γ ) = + ∞ , and the result holds, hene w e ma y assume that p γ b elongs to X . If p γ do es not b elong to H , as the losest p oin t pro jetion of γ on the geo desi line ] γ − , ξ [ is ] γ − , p γ [ , then γ do es not en ter H , and hene ipp H ( γ ) = ℓ H ( γ ) = 0 , and the result is pro v en. ξ p γ ∂ H x y γ − γ + Assume that p γ b elongs to H , and note that b y ompari- son and an easy h yp erb oli estimate, w e ha v e d ( p γ , γ ) ≤ log(1 + √ 2) . In partiular, if γ do es not en ter H , then 0 ≤ β H ( p γ ) ≤ d ( p γ , γ ) ≤ log(1 + √ 2) , and | ipp H ( γ ) − ℓ H ( γ ) | ≤ 2 log (1 + √ 2) , hene the result holds. Therefore w e ma y assume that γ en ters H at the p oin t x and exits H at the p oin t y . W e then ha v e ph H ( γ ) ≤ ipp H ( γ ) ≤ ph H ( γ ) + 2 d ( p γ , γ ) . Hene, ℓ H ( γ ) − 2 log (1+ √ 2) ≤ ipp H ( γ ) ≤ ℓ H ( γ ) +2 log (1+ √ 2) , and the result is pro v en. The on tin uit y of ipp H follo ws from the on tin uit y of the endp oin t maps, of the losest p oin t pro jetion maps and of β H : X ∪ { ξ } → [0 , + ∞ ] . T o pro v e the on tin uit y of ph H at a p oin t γ 0 of T 1 ξ 0 X , note that if γ 0 (+ ∞ ) = ξ , then ph H ( γ 0 ) = + ∞ , and the on tin uit y follo ws from the P enetration prop ert y ( i ) of ( H , ph H ) and the on tin uit y of ℓ H . Otherwise, γ 0 ∩ H is a ompat segmen t. If it is nonempt y , then if γ is lose enough to γ ′ , the argumen t in the pro of of Lemma 3.1 sho ws that the Hausdor distane b et w een γ ∩ H and γ 0 ∩ H is as small as w an ted. The result follo ws then sine β H is 1 -Lips hitz. If γ 0 do es not meet H , then if γ is lose enough to γ ′ , the argumen t in the pro of of Lemma 3.1 sho ws that γ also a v oids H , hene ph H ( γ ) = ℓ H ( γ ) = 0 .  P enetration maps in balls. If B is a ball of en ter x 0 and radius r 0 in X with ξ 0 / ∈ B , dene a 1 -Lips hitz map β B : X → [0 , + ∞ [ , alled the height map b y β B : x 7→ max { r 0 − d ( x 0 , x ) , 0 } , whose v alues are p ositiv e in the in terior of B , and 0 outside B . F or ev ery geo desi line γ in T 1 ξ 0 X , let p γ b e the losest p oin t to γ (+ ∞ ) on the geo desi segmen t or ra y b et w een ξ 0 and x 0 . The map ph B : T 1 ξ 0 X → [0 , 2 r 0 ] dened b y ph B ( γ ) = 2 su p t ∈ R β B ( γ ( t )) will b e alled the p enetr ation height map inside B . The map ipp B : T 1 ξ 0 X → [0 , 2 r 0 ] dened b y ipp B ( γ ) = 2 β B ( p γ ) will b e alled the inner-pr oje tion p enetr ation map inside B . 25 As ab o v e, the maps ph B , ipp B are on tin uous 2 log(1 + √ 2) -p enetration maps, and ph B has the Lips hitz prop ert y (ii) . F urthermore, k ph B − ipp B k ∞ ≤ 2 log (1 + √ 2) . In the pro of of the P enetration prop ert y of ipp B , note that if p γ = x 0 , then, b y omparison, d ( γ , x 0 ) ≤ log(1 + √ 2) , and the laim follo ws in this ase as ipp B ( γ ) = 2 r 0 . If a sequene of balls ( B i ) i ∈ N on v erges to an horoball H (for the Hausdor distane on ompat subsets of X ), then the maps ph B i , ipp B i on v erge, uniformly on ompat subsets of T 1 ξ 0 X , to ph H , ipp H resp etiv ely . P enetration maps in tubular neigh b orho o ds of totally geo desi subspaes. W e dene t w o funtions on T 1 ξ 0 X whi h desrib e the loseness of a geo desi line to a totally geo desi subspae L . If ξ 0 is in the b oundary at innit y of X , then these funtions are dened without referene to an ǫ -neigh b ourho o d of L . Ho w ev er, w e sho w that they are p enetration maps in the ǫ -neigh b ourho o d of L , with expliit onstan ts whi h dep end only on ǫ . Let ǫ > 0 , and let L b e a omplete totally geo desi subspae of X , with set of p oin ts at innit y ∂ ∞ L , su h that ξ 0 / ∈ N ǫ L ∪ ∂ ∞ L . F or ev ery γ in T 1 ξ 0 X , let γ − = ξ 0 and γ + = γ (+ ∞ ) , and let p γ ± b e the p oin t on L the losest to γ ± . p γ − γ + γ − p γ + L x q γ + q γ − y W e dene the fel low-tr avel ler p enetr ation map ftp L : T 1 ξ 0 X → [0 , + ∞ ] b y ftp L ( γ ) = d ( p γ − , p γ + ) , with the on v en tion that this distane is + ∞ if p γ + is in ∂ ∞ X . Lemma 3.4 L et ǫ > 0 , and let L b e a  omplete total ly ge o desi subsp a e of X suh that ξ 0 / ∈ N ǫ L ∪ ∂ ∞ L . The map ftp L is a  ontinuous (2 c ′ 1 ( ǫ ) + 2 ǫ ) -p enetr ation map in N ǫ L and k ftp L − bp N ǫ L k ∞ ≤ 2 ǫ . Pro of. The on tin uit y of ftp L follo ws from the on tin uit y of the pro jetion maps and of the endp oin t maps. Note that, for ev ery γ in T 1 ξ 0 X , the geo desi ra y from p γ ± to γ ± exits N ǫ L at the losest p oin t q γ ± on N ǫ L to γ ± . Hene, b y the triangular inequalit y , and as losest p oin t maps do not inrease distanes, w e ha v e 0 ≤ bp N ǫ L ( γ ) − ftp L ( γ ) ≤ 2 ǫ . Therefore the fat that ftp L ( γ ) satises the P enetration prop ert y ( i ) with κ = 2 c ′ 1 ( ǫ ) + 2 ǫ follo ws from Lemma 3.2 .  If L is one-dimensional and ξ 0 ∈ ∂ ∞ X − ∂ ∞ L , a natural p enetration map is dened using the rossratios of the endp oin ts of L and γ . Let ∂ 4 X b e the set of quadruples 26 ( a, b, c, d ) in ( ∂ ∞ X ) 4 su h that a 6 = b and c 6 = d . The r ossr atio [ a, b, c, d ] ∈ [ −∞ , + ∞ ] of a quadruple ( a, b, c, d ) in ∂ 4 X is dened as follo ws (see for instane [ Ota, Bou, P au ℄). If a t , b t , c t , d t are an y geo desi ra ys on v erging to a, b, c, d resp etiv ely , then [ a, b, c, d ] = 1 2 lim t → + ∞ d ( a t , c t ) − d ( c t , b t ) + d ( b t , d t ) − d ( d t , a t ) . Note that the order on v en tions dier in the referenes, w e are using the ones of [Bou, HP2 ℄), and that our rossratio is the logarithm of the rossratio used in [Bou ℄. Let us giv e other form ulae for the rossratio. The visual distan e of t w o p oin ts a and b in ∂ ∞ X with resp et to x 0 is d x 0 ( a, b ) = li m t →∞ e − 1 2  d ( x 0 ,a t )+ d ( x 0 ,b t ) − d ( a t ,b t )  . If ξ ∈ ∂ ∞ X , if H is a horosphere en tered at ξ , and a, b are p oin ts in ∂ ∞ X − { ξ } , and t 7→ x t is a geo desi ra y with x 0 ∈ H whi h on v erges to ξ , the Hamenstädt distan e (dened in [Ham℄, [HP2 , App endix℄) of a and b in ∂ ∞ X − { ξ } normalized with resp et to H is d H ( a, b ) = lim t →∞ e t d x t ( a, b ) . Note that if H ′ is another horosphere en tered at ξ , then there exists a onstan t c > 0 su h that d H ′ = c d H . In partiular, for ev ery ξ ′ ∈ ∂ ∞ X − { ξ } and r > 0 , the sphere of en ter ξ ′ and radius r for d H oinides with the sphere of en ter ξ ′ and radius r for d H ′ . It is easy to see that for an y x 0 ∈ X and an y horoball H , w e ha v e for ev ery ( a, b, c, d ) ∈ ∂ 4 X [ a, b, c, d ] = log d x 0 ( a, c ) d x 0 ( c, b ) d x 0 ( b, d ) d x 0 ( d, a ) = log d H ( a, c ) d H ( c, b ) d H ( b, d ) d H ( d, a ) , if, in the seond equation, a, b, c, d are in ∂ ∞ X − { ξ } . Note that ea h expression in the ab o v e t w o equalities is −∞ if a = c or b = d , and + ∞ if c = b or a = d . If the p oin ts ξ and a oinide, the expression of the rossratio simplies to [ ξ , b, c, d ] = log d H ( b, d ) d H ( c, b ) . The rossratio is on tin uous on ∂ 4 X , it is in v arian t under the diagonal ation of the isometry group of Γ , and it has the follo wing symmetries [ c, d, a, b ] = [ a, b, c, d ] a n d [ a, b, d, c ] = [ b, a, c, d ] = − [ a, b, c, d ] . If X = H n R and ξ is the p oin t at innit y ∞ in the upp er halfspae mo del of H n R , then the Hamenstädt distane oinides with a onstan t m ultiple of the Eulidean distane of ∂ ∞ H n R − {∞} = R n − 1 (see for instane [HP3 ℄). In partiular, if n = 2 , then our rossratio is the logarithm of the mo dulus of the lassial rossratio of four p oin ts in C ∪ {∞} . If ξ 0 ∈ ∂ ∞ X − ∂ ∞ L , w e dene the r ossr atio p enetr ation map crp L : T 1 ξ 0 X → [0 , + ∞ ] as follo ws. Let γ b e a geo desi line starting at γ − = ξ 0 , and ending at γ + ∈ ∂ X . Let L 1 , L 2 b e the endp oin ts of L . Set crp L ( γ ) = m ax  0 , [ γ − , L 1 , γ + , L 2 ] , [ γ − , L 2 , γ + , L 1 ]  , 27 if γ + 6 = L 1 , L 2 , and crp L ( γ ) = + ∞ otherwise. The map crp L is learly on tin uous, in partiular sine [ a, b, c, d ] tends to 0 when a, b, d are pairwise distint and c tends to d , and is indep endan t of the ordering L 1 , L 2 of the endp oin ts of L . If H is a horosphere en tered at ξ 0 , then [ ξ 0 , L 1 , γ + , L 2 ] = log d H ( L 1 , L 2 ) d H ( γ + , L 1 ) , and the lev el sets for crp L ha v e a simple form: [ ξ 0 , L 1 , γ + , L 2 ] = c if and only if γ + is on the sphere of radius e − c d H ( L 1 , L 2 ) en tered at L 1 with resp et to the Hamenstädt metri. Th us, in partiular, the b oundary of the zero set of crp L is the b oundary of the union of the t w o balls of radius d H ( L 1 , L 2 ) en tered at L 1 and L 2 . F urthermore, if c > log 2 , then the lev el set crp L − 1 ( c ) is the union of t w o spheres for the Hamenstädt distane d H of en ters L 1 and L 2 and radius e − c d H ( L 1 , L 2 ) . These t w o spheres are disjoin t b y the triangle inequalit y . Ea h of them separates ξ 0 from exatly one of the endp oin ts of L . W e will use this in the pro of of Lemma 3.9 . 0 2 log 2 log 2 1 2 log 2 d H ( L 1 , L 2 ) L 1 L 2 Note that if X is a negativ ely urv ed symmetri spae, then the spheres and balls of the Hamenstädt distane are top ologial spheres and balls in the top ologial sphere ∂ ∞ X (see [HP3℄ if X = H n R and [HP4 ℄ if X = H n C ). W e don't kno w (and in fat w e doubt it) whether this alw a ys holds in the general v ariable urv ature ase. Lemma 3.5 L et ( a, b, c, d ) ∈ ∂ 4 X . If b = d , we dene by  onvention p = q = b and d ( p, q ) = 0 . Otherwise, let p and q b e the losest p oints on [ b, d ] of a and c r esp e tively. (1) If b, q , p, d ar e in this or der on [ b, d ] and d ( p, q ) ≥ c ′ 1 ( ∞ ) , then   [ a, b, c, d ] − d ( p, q )   ≤ 2 c ′ 1 ( ∞ ) . (2) If b, p, q , d ar e in this or der on ] b, d [ and d ( p, q ) ≥ c ′ 1 ( ∞ ) , then [ a, b, c, d ] ≤ c ′ 1 ( ∞ ) . (3) If d ( p, q ) ≤ c ′ 1 ( ∞ ) , then [ a, b, c, d ] ≤ 2 c ′ 1 ( ∞ ) . Pro of. If a = d or c = b , then p = d or q = b , hene w e are in ase (1) and [ a, b, c, d ] = d ( p, q ) = ∞ , whi h pro v es the result. If a = c or b = d , then p = q , w e are in ase (3) and [ a, b, c, d ] = −∞ , whi h pro v es the result. Hene w e ma y assume that a, b, c, d are pairwise disjoin t. 28 Let a t , b t , c t , d t b e geo desi ra ys on v erging to resp etiv ely a, b, c, d as t → ∞ , and let p t and q t b e the losest p oin ts to a t and c t resp etiv ely on [ b t , d t ] . Let p ′ ∈ [ a t , d t ] and p ′′ ∈ [ a t , c t ] b e the losest p oin ts to p t on [ a t , d t ] and [ a t , c t ] , and let q ′ ∈ [ b t , c t ] and q ′′ ∈ [ a t , c t ] b e the losest p oin ts to q t on [ b t , c t ] and [ a t , c t ] . Reall that b y an easy omparison argumen t, for pairwise distint p oin ts u, v , w in X ∪ ∂ ∞ X , if r is the losest p oin t to w on ] u, v [ , then r is at distane less than δ = log (1+ √ 2) from a p oin t on ] u, w [ . W e will apply this remarkto r = p t and r = q t . Reall also that c ′ 1 ( ∞ ) = 2 δ . Case 2 Case 3 Case 1 b t d t p t q t q ′ c t a t q ′′ p ′ b t p t q t q ′ p ′′ a t c t q ′′ d t p t q t p ′′ c t a t q ′ q ′′ p ′ b t d t p ′′ p ′ Case (1). If t is big enough, then the p oin ts b t , q t , p t , d t are in this order on [ b t , d t ] . Using the triangle inequalit y on d ( a t , c t ) and d ( b t , d t ) , and inserting the p oin ts p ′ and q ′ , w e ha v e d ( a t , c t ) − d ( c t , b t ) + d ( b t , d t ) − d ( d t , a t ) ≤ d ( a t , p t ) + d ( p t , q t ) + d ( q t , c t ) − d ( c t , q t ) − d ( q t , b t ) + 2 d ( q ′ , q t ) + d ( b t , q t ) + d ( q t , p t ) + d ( p t , d t ) − d ( d t , p t ) − d ( p t , a t ) + 2 d ( p ′ , p t ) ≤ 2 d ( p t , q t ) + 4 δ . By omparison and a standard argumen t on h yp erb oli quadrilaterals with three righ t angles (see [Bea , page 157℄), for ev ery ǫ > 0 , if t is big enough, w e ha v e that d ( q t , q ′′ ) ≤ 2 δ + ǫ/ 4 , as d ( p t , q t ) → d ( p, q ) ≥ c ′ 1 ( ∞ ) . If w e insert the p oin ts p ′′ and q ′′ , w e get, as ab o v e, d ( a t , c t ) − d ( c t , b t ) + d ( b t , d t ) − d ( d t , a t ) ≥ d ( a t , p t ) + d ( p t , q t ) + d ( q t , c t ) − 2 d ( p t , p ′′ ) − 2 d ( q t , q ′′ ) − d ( c t , q t ) − d ( q t , b t ) + d ( b t , q t ) + d ( q t , p t ) + d ( p t , d t ) − d ( d t , p t ) − d ( p t , a t ) ≥ 2 d ( p t , q t ) − 8 δ + ǫ . Case (2). The pro of is almost iden tial to the one of the upp er b ound in the rst inequalit y in Case 1. The dieren t order of the p oin ts p t and q t no w auses anellations: d ( a t , c t ) − d ( c t , b t ) + d ( b t , d t ) − d ( d t , a t ) ≤ d ( a t , p t ) + d ( p t , q t ) + d ( q t , c t ) − d ( c t , q t ) − d ( q t , p t ) − d ( p t , b t ) + 2 d ( q ′ , q t ) + d ( b t , p t ) + d ( p t , q t ) + d ( q t , d t ) − d ( d t , q t ) − d ( q t , p t ) − d ( p t , a t ) + 2 d ( p ′ , p t ) ≤ 4 δ . Case (3). Let ǫ > 0 . By taking t big enough, w e an assume that d ( p t , q t ) ≤ 2 δ + ǫ . Inserting the p oin ts p ′′ and q ′′ and using the fat that losest p oin t maps do not inrease 29 distanes, w e ha v e d ( p ′′ , q ′′ ) ≤ 2 δ + ǫ , d ( c t , q t ) ≥ d ( c t , q ′′ ) and d ( a t , p t ) ≥ d ( a t , p ′′ ) . Th us, as in the ases ab o v e, d ( a t , c t ) − d ( c t , b t ) + d ( b t , d t ) − d ( d t , a t ) ≤ d ( a t , p ′′ ) + d ( p ′′ , q ′′ ) + d ( q ′′ , c t ) − d ( c t , q t ) − d ( q t , b t ) + 2 d ( q ′ , q t ) + d ( b t , q t ) + d ( q t , p t ) + d ( p t , d t ) − d ( d t , p t ) − d ( p t , a t ) + 2 d ( p ′ , p t ) ≤ 2 8 δ + 2 ǫ . As this holds for an y ǫ > 0 , the result follo ws.  Lemma 3.6 L et ǫ > 0 , let L b e a ge o desi line in X , and assume that ξ 0 ∈ ∂ ∞ X − ∂ ∞ L . The map crp L is a  ontinuous (2 c ′ 1 ( ǫ ) + 2 c ′ 1 ( ∞ ) + 2 ǫ ) -p enetr ation map in the ǫ - onvex set N ǫ L and k crp L − ftp L k ∞ ≤ 2 c ′ 1 ( ∞ ) . Pro of. Let γ ∈ T 1 ξ 0 X , let γ − = ξ 0 and γ + b e the endp oin ts of γ , and L 1 and L 2 b e the endp oin ts of L . Let p and q b e the losest p oin ts to γ − and γ + on L resp etiv ely . If d ( p, q ) ≤ c ′ 1 ( ∞ ) , then Lemma 3.5 implies that 0 ≤ crp L ( γ ) ≤ 2 c ′ 1 ( ∞ ) , and th us | crp L ( γ ) − ftp L ( γ ) | ≤ 2 c ′ 1 ( ∞ ) . If d ( p, q ) > c ′ 1 ( ∞ ) , then up to renaming the endp oin ts of L , w e ha v e [ γ − , L 2 , γ + , L 1 ] ≤ c ′ 1 ( ∞ ) and − 2 c ′ 1 ( ∞ ) + ftp L ( γ ) ≤ [ γ − , L 1 , γ + , L 2 ] ≤ 2 c ′ 1 ( ∞ ) + f tp L ( γ ) , whi h implies the result.  Remark. The p enetration maps an b e dened for an y xed starting p oin t whi h is outside the ǫ -on v ex set C , exept for crp L , and its b oundary at innit y . Th us, the p enetration maps ℓ C , bp C , ph H , ipp H , ph B , ipp B , ftp L onsidered in this setion are all restritions to T 1 ξ 0 X of maps dened, and on tin uous (as an insp etion of the ab o v e pro of sho ws) on S ξ / ∈ C ′ ∪ ∂ ∞ C ′ T 1 ξ X ⊂ T 1 X with C ′ resp etiv ely C, C , H , H , B , B , N ǫ L . The p enetration map crp L is dened and on tin uous on S ξ ∈ ∂ ∞ X − ∂ ∞ C ′ T 1 ξ X . This p oin t of view is used in ases (3) and (4) of Prop osition 3.7 b elo w, and will b e useful to apply Corollary 4.11. 3.2 Presribing the p enetration In Setion 4, w e will use the follo wing op eration rep eatedly: a geo desi ra y or line γ starting from a giv en p oin t ξ 0 is giv en that p enetrates t w o ǫ -on v ex sets C and C ′ with p enetration maps f and f ′ , rst en tering C with f ( γ ) = h , and then C ′ with f ′ ( γ ) ≥ h ′ . W e will need to pi k a new geo desi ra y or line γ ′ starting from ξ 0 whi h in tersets C b efore C ′ , for whi h w e still ha v e f ( γ ′ ) = h , and for whi h w e no w ha v e the equalit y f ′ ( γ ′ ) = h ′ . In the follo wing result, w e sho w that this op eration is p ossible in a n um b er of geometri ases. These ases will b e used in Setion 5 for v arious appliations. Prop osition 3.7 L et X b e a  omplete, simply  onne te d R iemannian manifold with se - tional urvatur e at most − 1 and dimension at le ast 3 . L et ǫ > 0 and δ , h, h ′ ≥ 0 . L et C and C ′ b e ǫ - onvex subsets of X , and ξ 0 ∈ ( X ∪ ∂ ∞ X ) − ( C ∪ ∂ ∞ C ) . L et f and f ′ b e maps T 1 ξ 0 X → [0 , + ∞ ] , with f ′  ontinuous and κ ′ = k f ′ − ℓ C ′ k ∞ < + ∞ . Consider the fol lowing  ases: 30 (1) C is a hor ob al l with diam( C ∩ C ′ ) ≤ δ ; f is either the p enetr ation height map ph C or the inner-pr oje tion p enetr ation map ipp C ; h ≥ h min = 2 c ′ 1 ( ǫ ) + 2 δ + k f − ph C k ∞ and h ′ ≥ h min 0 = κ ′ + 2 δ ; if C ′ is also a hor ob al l we may take ǫ = + ∞ in the denition of h min . (2) C is a b al l of r adius R ( ≥ ǫ ) with diam( C ∩ C ′ ) ≤ δ ; f is either the p enetr ation height map ph C or the inner-pr oje tion p enetr ation map ipp C ; h min = 2 c ′ 1 ( ǫ ) + 2 δ + k f − ph C k ∞ ≤ h ≤ 2 R − 2 c ′ 1 ( ǫ ) − k f − ph C k ∞ = h max and h ′ ≥ h min 0 = κ ′ + 2 δ ; (3) C is the ǫ -neighb ourho o d of a  omplete total ly ge o desi subsp a e L of dimension at le ast 2 , with diam( C ∩ C ′ ) ≤ δ ; either f = ℓ C and X has  onstant urvatur e, or f is the fel low-tr avel ler p enetr ation map ftp L ; h ≥ h min = 4 c ′ 1 ( ǫ ) + 2 ǫ + δ + k f − ftp L k ∞ and h ′ > h min 0 = κ ′ + δ ; (4) • C is the ǫ -neighb ourho o d of a ge o desi line L ; • h ≥ h min = 4 c ′ 1 ( ǫ ) + 2 ǫ + δ + k f − ftp L k ∞ ; • either f = ℓ C and X has  onstant urvatur e, or f = crp L , ξ 0 ∈ ∂ ∞ X and the metri spher es of the Hamenstädt distan e on ∂ ∞ X − { ξ 0 } ar e top olo gi al spher es, or f is the fel low-tr avel ler p enetr ation map ftp L ; • either C ′ is any ǫ - onvex subset that do es not me et C (in whih  ase δ = 0 ) and h ′ > h min 0 = κ ′ , or C ′ is the ǫ -neighb ourho o d of a total ly ge o desi subsp a e with  o dimension at le ast two suh that diam( C ∩ C ′ ) ≤ δ and h ′ ≥ h min 0 = 3 c ′ 1 ( ǫ ) + 3 ǫ + δ + k f ′ − ftp L ′ k ∞ . Assume that one of the ab ove  ases holds. If ther e exists a ge o desi r ay or line γ starting fr om ξ 0 whih me ets rst C and then C ′ with f ( γ ) = h and f ′ ( γ ) ≥ h ′ , then ther e exists a ge o desi r ay or line γ starting fr om ξ 0 whih me ets rst C and then C ′ with f ( γ ) = h and f ′ ( γ ) = h ′ . Pro of. Let γ b e as in the statemen t, and x (resp. y ) b e the p oin t where γ en ters (resp. ex- its) C (with y in X as f ( γ ) = h < + ∞ ). Let x ′ (resp. y ′ ) b e the p oin t where γ en ters (resp. exits) C ′ (with x ′ ∈ X but p ossibly with y ′ at innit y). By on v exit y , ξ 0 / ∈ C ′ ∪ ∂ ∞ C ′ . F or ev ery h ≥ 0 , w e dene A as the set of p oin ts α (+ ∞ ) where α ∈ T 1 ξ 0 X satises f ( α ) = h . Let A 0 b e the arwise onneted omp onen t of A on taining γ (+ ∞ ) . By onsidering the v arious ases, w e will pro v e b elo w the follo wing t w o laims : a) ev ery geo desi ra y or line, starting from ξ 0 and meeting C ′ , rst meets C and then C ′ ; 31 b) there exists a geo desi ra y or line γ 0 starting from ξ 0 with γ 0 (+ ∞ ) b elonging to A 0 , and f ′ ( γ 0 ) ≤ h min 0 . As f ′ is on tin uous and A 0 is arwise onneted, the in termediate v alue theorem implies the existene of a geo desi γ with the desired prop erties, and Prop osition 3.7 is pro v en. Case (1) . Let κ = k f − ph C k ∞ . Let ξ b e the p oin t at innit y of C , whi h is dieren t from ξ 0 , and let p ξ b e the losest p oin t to ξ on γ . As f ( γ ) = h > 0 , the p oin t p ξ b elongs to the in terior of the horoball C . Let γ ξ b e the geo desi ra y or line starting from ξ 0 with γ ξ (+ ∞ ) = ξ . x ′ ξ 0 p ξ u ′ C [ δ ] y ξ γ ξ x C C ′ y ′ γ W e start b y pro ving the (stronger) rst laim that ev ery geo desi ra y or line starting from ξ 0 and meeting C ′ meets C [ δ ] rst (hene meets C b efore C ′ ). Note that d ( y , p ξ ) ≥ d ( p ξ , ∂ C ) = ph C ( γ ) 2 ≥ f ( γ ) − κ 2 = h − κ 2 ≥ h min − κ 2 = c ′ 1 ( ǫ ) + δ > δ . As f ′ ( γ ) ≥ h ′ ≥ h min 0 = κ ′ + 2 δ , w e ha v e ℓ C ′ ( γ ) ≥ f ′ ( γ ) − κ ′ > δ , unless ℓ C ′ ( γ ) = δ = 0 . Note that γ ∩ C ′ is not on tained in the geo desi segmen t [ x, y ] . Otherwise, this w ould on tradit the assumption that diam( C ∩ C ′ ) ≤ δ when ℓ C ′ ( γ ) > δ . When ℓ C ′ ( γ ) = δ = 0 , as γ meets C ′ , the segmen t γ ∩ C ′ w ould b e redued to a p oin t b y the on v exit y of C ′ , whi h w ould b e { x } or { y } (as C ′ is not a singleton). But then the tangen t v etor of γ at x or its opp osite at y w ould b oth en ter stritly C and b e tangen t to C ′ , whi h on tradits the fat that δ = 0 . As γ meets C b efore C ′ , this implies in partiular that the geo desi ra y [ y , γ (+ ∞ )[ meets C ′ , and that the p oin t p ξ b elongs to ] x ′ , ξ 0 ] : otherwise C ∩ C ′ w ould on tain a segmen t of length at least d ( p ξ , y ) > δ , whi h is imp ossible. Hene b y on v exit y , an y geo desi ra y or line, starting from ξ 0 and meeting B ( x ′ , c ′ 1 ( ǫ )) , rst meets B ( p ξ , c ′ 1 ( ǫ )) . By Lemma 2.5, ev ery geo desi ra y or line, starting from ξ 0 and meeting C ′ , meets the ball B ( x ′ , c ′ 1 ( ǫ )) b efore en tering C ′ . This pro v es the rst laim, as the ball B ( p ξ , c ′ 1 ( ǫ )) is on tained in C [ δ ] , sine d ( p ξ , ∂ C ) ≥ c ′ 1 ( ǫ ) + δ , as seen ab o v e. Let us pro v e no w the (stronger) seond laim that there exists a geo desi ra y or line γ 0 starting from ξ 0 with γ 0 (+ ∞ ) b elonging to A 0 , and a v oiding the in terior of C ′ , whi h implies the result, as then f ′ ( γ 0 ) ≤ ℓ C ′ ( γ 0 ) + κ ′ ≤ h min 0 . 32 The subspae A of ∂ ∞ X is a o dimension 1 top ologial submanifold of the top ologial sphere ∂ ∞ X , whi h is homeomorphi to the sphere n − 2 , hene is arwise onneted. Indeed, if f = ph C , then A is the subset of endp oin ts of the geo desi ra ys or lines starting from ξ 0 that are tangen t to ∂  C [ h/ 2]  . If f = ipp C , the subset A is the preimage of a p oin t in ] ξ 0 , ξ [ b y the losest p oin t map from ∂ ∞ X to [ ξ 0 , ξ ] , whi h is, o v er ] ξ 0 , ξ [ , a trivial top ologial bundle with b ers homeomorphi to n − 2 . Note that f is on tin uous, f ( γ ξ ) = ∞ > h , and f ( α ) = 0 if α is a geo desi ra y or line starting from ξ 0 with α (+ ∞ ) lose enough to γ ( − ∞ ) . Therefore A separates γ ( − ∞ ) and γ ξ (+ ∞ ) , as the onneted omp onen ts of ∂ ∞ X − A are arwise onneted. If the (stronger) seond laim is not true, then the top ologial sphere A 0 = A of dimension n − 2 is on tained in the in terior of the shado w ξ 0 C ′ . As ξ 0 / ∈ C ′ ∪ ∂ ∞ C ′ , this shado w is homeomorphi to a ball of dimension n − 1 . Th us, b y Jordan's theorem, one of the t w o onneted omp onen ts of ∂ ∞ X − A is on tained in the in terior of ξ 0 C ′ . As γ ( − ∞ ) do es not b elong to ξ 0 C ′ and A separates γ ( − ∞ ) and γ ξ (+ ∞ ) , this implies that γ ξ (+ ∞ ) b elongs to the in terior of ξ 0 C ′ . Hene γ ξ meets the in terior of C ′ . Therefore, b y the rst laim, the geo desi ra y or line γ ξ meets C [ δ ] b efore meeting C ′ . Let u ′ b e the en tering p oin t of γ ξ in C ′ . As ξ is the p oin t at innit y of C [ δ ] , the p oin ts ξ 0 , u ′ , ξ are in this order on γ ξ . Hene b y on v exit y , this implies that u ′ b elongs to C [ δ ] . As f ′ ( γ ) ≥ h ′ ≥ h min 0 = κ ′ + 2 δ , w e ha v e d ( x ′ , y ′ ) = ℓ C ′ ( γ ) ≥ f ′ ( γ ) − κ ′ ≥ 2 δ . Hene b y the triangular inequalit y , one of the t w o distanes d ( u ′ , x ′ ) , d ( u ′ , y ′ ) is at least δ , and b y the strit on v exit y of the distane, is stritly bigger than δ (as u ′ do es not b elong to γ (as γ 6 = γ ξ ). Hene, if u ′′ is a p oin t lose enough to u in ] u, ξ [ , then u ′′ b elongs to the in terior of C ′ and to the in terior of C [ δ ] , and is at distane stritly greater than δ from either x ′ or y ′ . Therefore, some geo desi segmen t of length stritly bigger than δ is on tained in the in tersetion C ∩ C ′ . This on tradits the assumption that diam( C ∩ C ′ ) ≤ δ . Case (2) . The pro of is ompletely similar to Case (1). Let no w C = B ( z , R ) , let p z b e the p oin t of γ the losest to z , and let γ z b e the geo desi ra y or line starting from ξ 0 and passing through z . Note that R > h max / 2 ≥ h min / 2 ≥ δ . W e only ha v e to replae ξ b y z , p ξ b y p z and γ ξ b y γ z , and to replae t w o argumen ts in the ab o v e pro of, the one in order to sho w that A separates γ ( − ∞ ) from γ z (+ ∞ ) , and the one in order to sho w that ξ 0 , u ′ , z are in this order on γ z , where u ′ is the en tering p oin t of γ z in C ′ . T o pro v e that A separates γ ( − ∞ ) from γ z (+ ∞ ) , w e simply use no w that f ( γ z ) = 2 R > h max ≥ h instead of f ( γ ξ ) = ∞ > h . Let us pro v e that ξ 0 , u ′ , z are in this order on γ z . W e ha v e, with κ = k f − ph C k ∞ , d ( z , p z ) = R − ph C ( γ ) 2 ≥ R − f ( γ ) + κ 2 ≥ R − h max + κ 2 = c ′ 1 ( ǫ ) . By Lemma 1.3, w e ha v e d ( x ′ , u ′ ) ≤ c ′ 1 ( ǫ ) . As γ z meets the in terior of C ′ , b y the same argumen t as in Case 1, w e ev en ha v e d ( u ′ , γ ) < c ′ 1 ( ǫ ) . Hene b y strit on v exit y , w e do ha v e u ′ ∈ ] z , ξ 0 [ . The rest of the argumen t in the pro of of Case (1 ) is un hanged. Before studying the last t w o ases, w e start b y pro ving t w o lemmas. The rst one implies the rst of the t w o laims w e need to pro v e in Cases (3 ), (4 ), and the seond one giv es the top ologial information on A that w e will need in these last t w o ases. 33 Lemma 3.8 L et L b e a  omplete total ly ge o desi subsp a e with dimension at le ast 1 , ǫ > 0 , C = N ǫ L , ξ 0 ∈ ( X ∪ ∂ ∞ X ) − ( C ∪ ∂ ∞ L ), and C ′ b e an ǫ - onvex subset of X suh that diam( C ∩ C ′ ) ≤ δ . L et f , f ′ : T 1 ξ 0 X → [0 , + ∞ ] b e maps suh that κ = k f − ftp L k ∞ < + ∞ , κ ′ = k f ′ − ℓ C ′ k ∞ < + ∞ . L et γ b e a ge o desi r ay or line starting fr om ξ 0 , entering C b efor e entering C ′ , suh that 4 c ′ 1 ( ǫ ) + 2 ǫ + δ + κ ≤ f ( γ ) ≤ + ∞ and f ′ ( γ ) > δ + κ ′ . If e γ is a ge o desi r ay or line starting fr om ξ 0 whih me ets C ′ , then e γ me ets the interior of C b efor e me eting C ′ . Pro of. Note that ξ 0 / ∈ C ′ ∪ ∂ ∞ C ′ , b y on v exit y and the assumptions on γ , as ξ / ∈ C ∪ ∂ ∞ C . Let L 0 b e the geo desi line passing through the losest p oin ts p ξ 0 , p γ (+ ∞ ) on L of ξ 0 , γ (+ ∞ ) , resp etiv ely . Note that d ( p ξ 0 , p γ (+ ∞ ) ) = ftp L ( γ ) ≥ f ( γ ) − κ ≥ 4 c ′ 1 ( ǫ ) + 2 ǫ + δ > 0 . Hene, b y Lemma 3.4 , and as ftp L 0 ( γ ) = f tp L ( γ ) , w e ha v e ℓ N ǫ L 0 ( γ ) ≥ f tp L 0 ( γ ) − 2 c ′ 1 ( ǫ ) − 2 ǫ > 0 . In partiular, γ en ters N ǫ L 0 at a p oin t x 0 and exits it at a p oin t y 0 in X (as γ (+ ∞ ) / ∈ ∂ ∞ L ). Let u 7→ p u b e the losest p oin t map from X ∪ ∂ ∞ X on to L 0 ∪ ∂ ∞ L 0 . Reall that this map do es not inrease the distanes (and ev en dereases them, unless the t w o p oin ts under onsideration are on L 0 ), and that it pr eserves b etwe enness , that is, if u ′′ ∈ [ u, u ′ ] , then p u ′′ ∈ [ p u , p u ′ ] . Let x ′ (resp. e x ′ ) b e the p oin t where γ (resp. e γ ) en ters C ′ , and q ξ 0 and q γ (+ ∞ ) b e the losest p oin t to ξ 0 and γ (+ ∞ ) resp etiv ely on N ǫ L 0 . γ (+ ∞ ) p ξ 0 ξ 0 p γ (+ ∞ ) e γ (+ ∞ ) p e γ (+ ∞ ) p e x ′ p x ′ p y 0 y 0 q ξ 0 q γ (+ ∞ ) L 0 ⊂ L e γ x ′ x 0 e x ′ γ C ′ Reall that b y Lemma 2.5, the distanes d ( e x ′ , x ′ ) , d ( x 0 , q ξ 0 ) , d ( y 0 , q γ (+ ∞ ) ) are at most c ′ 1 ( ǫ ) . Note that e x ′ ∈ [ ξ 0 , e γ (+ ∞ )] . Hene, as b et w eennes is preserv ed, ftp L 0 ( e γ ) = d ( p ξ 0 , p e γ (+ ∞ ) ) ≥ d ( p ξ 0 , p e x ′ ) ≥ d ( p ξ 0 , p x ′ ) − d ( p x ′ , p e x ′ ) ≥ d ( p ξ 0 , p x ′ ) − d ( x ′ , e x ′ ) ≥ d ( p ξ 0 , p x ′ ) − c ′ 1 ( ǫ ) . Note that d ( p ξ 0 , p x ′ ) ≥ d ( p ξ 0 , p y 0 ) when ξ 0 , y 0 , x ′ are in this order on γ . When ξ 0 , y 0 , x ′ are not in this order on γ , as γ en ters in C b efore C ′ , as ℓ C ′ ( γ ) ≥ f ′ ( γ ) − κ ′ > δ and as diam( C ∩ C ′ ) ≤ δ , w e ha v e d ( x ′ , y 0 ) ≤ δ ; hene d ( p ξ 0 , p x ′ ) ≥ d ( p ξ 0 , p y 0 ) − d ( p y 0 , p x ′ ) ≥ d ( p ξ 0 , p y 0 ) − d ( y 0 , x ′ ) ≥ d ( p ξ 0 , p y 0 ) − δ . 34 Therefore, in b oth ases, as y 0 ∈ [ ξ 0 , γ (+ ∞ )] and u 7→ p u preserv es the b et w eenness, and sine p γ (+ ∞ ) = p q γ (+ ∞ ) , w e ha v e ftp L 0 ( e γ ) ≥ d ( p ξ 0 , p e x ′ ) ≥ d ( p ξ 0 , p y 0 ) − δ − c ′ 1 ( ǫ ) ≥ d ( p ξ 0 , p γ (+ ∞ ) ) − d ( p q γ (+ ∞ ) , p y 0 ) − c ′ 1 ( ǫ ) − δ > ftp L ( γ ) − d ( q γ (+ ∞ ) , y 0 ) − c ′ 1 ( ǫ ) − δ ≥ f tp L ( γ ) − 2 c ′ 1 ( ǫ ) − δ ≥ 2 c ′ 1 ( ǫ ) + 2 ǫ . By Lemma 3.4, w e hene ha v e ℓ N ǫ L ( e γ ) ≥ ℓ N ǫ L 0 ( e γ ) ≥ ftp L 0 ( e γ ) − 2 c ′ 1 ( ǫ ) − 2 ǫ > 0 . In partiular, e γ do es en ter the in terior of C , at a p oin t e x . Note that the geo desi from ξ 0 through p ξ 0 en ters C at q ξ 0 . No w b y absurd, if e γ en ters the in terior of C after it en ters C ′ , then e x ′ ∈ [ ξ 0 , e x ] , so that c ′ 1 ( ǫ ) ≥ d ( q ξ 0 , e x ) ≥ d ( p ξ 0 , p e x ) ≥ d ( p ξ 0 , p e x ′ ) > 2 c ′ 1 ( ǫ ) + 2 ǫ , as seen ab o v e, a on tradition.  Lemma 3.9 L et X b e a  omplete, simply  onne te d R iemannian manifold with se tional urvatur e at most − 1 and dimension at le ast 3 . L et ǫ, h > 0 . L et L b e a  omplete total ly ge o desi submanifold with dimension at le ast 1 and ξ 0 ∈ ( X ∪ ∂ ∞ X ) − ( N ǫ L ∪ ∂ ∞ L ) . Assume either that (1) f = ℓ N ǫ L , X has  onstant urvatur e and h ∈ [4 c ′ 1 ( ǫ ) + 2 ǫ, + ∞ [ , or (2) f = crp L , ξ 0 ∈ ∂ ∞ X , dim L = 1 , h ∈ ]log 2 , + ∞ [ , and the metri spher es of the Hamenstädt distan e on ∂ ∞ X − { ξ 0 } ar e top olo gi al spher es, or (3) f = ftp L . Then A = { α (+ ∞ ) : α ∈ T 1 ξ 0 X, f ( α ) = h } is a  o dimension 1 top olo gi al submanifold of the top olo gi al spher e ∂ ∞ X , whih is home- omorphi to the torus dim L − 1 × codim L − 1 . F urthermor e, (a) if dim L = 1 , then A has two ar wise  onne te d  omp onents, home omorphi to a spher e of dimension n − 2 . If f = crp L or if h > c ′ 1 ( ǫ ) , then e ah of them sep ar ates γ ( − ∞ ) and exatly one of the two p oints at innity of L , for every ge o desi r ay or line γ starting fr om ξ 0 if f = crp L , and for those me eting N ǫ L if f 6 = crp L . (b) if co dim L = 1 , then A has two ar wise  onne te d  omp onents, home omorphi to a spher e of dimension n − 2 , sep ar ate d by ∂ ∞ L . () if dim L ≥ 2 and co dim L ≥ 2 , then A is ar wise  onne te d. In  ases (b) and (), for every  omp onent A ′ of A , for every ge o desi r ay ρ in L with ρ (0) the losest p oint to ξ 0 on L , ther e exists η ∈ A ′ suh that ρ ( h ) is at distan e at most k f − ftp L k ∞ fr om the losest p oint to η on ρ . 35 Pro of. Let π L : X ∪ ∂ ∞ X → L ∪ ∂ ∞ L b e the losest p oin t map, p 0 = π L ( ξ 0 ) and S 0 = π − 1 L ( p 0 ) ∩ ∂ ∞ X . Assume rst that f = ftp L . As h > 0 , the subspae A of ∂ ∞ X is the preimage of the sphere (of dimension dim L − 1 ) of en ter p 0 and radius h in L , b y π L . As π L : ∂ ∞ X \ ∂ ∞ L → L is a trivial top ologial bundle whose b ers are spheres of dimension co dim L − 1 , the top ologial struture of A is immediate. The nal statemen t on (b) and () is trivial as, b y denition, ρ ( h ) is the losest p oin t to some p oin t in A ′ . If dim L = 1 and h > c ′ 1 ( ǫ ) , if γ ∈ T 1 ξ 0 X meets N ǫ L , then b y Lemma 2.5 and b y on v exit y , d ( π L ( γ ( −∞ )) , p 0 ) ≤ c ′ 1 ( ǫ ) < h . Hene, the separation statemen t in (a) follo ws. Assume no w that f = crp L , and that the h yp otheses of (2) are satised. The result in this ase follo ws from the disussion b efore Lemma 3.5. Assume no w that f = ℓ N ǫ L , X has onstan t urv ature, and h ∈ [4 c ′ 1 ( ǫ ) + 2 ǫ, + ∞ [ . Using normal o ordinates along L , the top ologial sphere ∂ ∞ X is homeomorphi to the top ologial join of the spheres ∂ ∞ L of dimension dim L − 1 and S 0 of dimension co dim L − 1 S 0 ∨ ∂ ∞ L =  S 0 × [0 , + ∞ ] × ∂ ∞ L  / ∼ , where ∼ is the equiv alene relation generated b y ( a, 0 , b ) ∼ ( a, 0 , b ′ ) and ( a, + ∞ , b ) ∼ ( a ′ , + ∞ , b ) , for ev ery a, a ′ in S 0 and b, b ′ in ∂ ∞ L . W e denote b y [ a, t, b ] the equiv alene lass of ( a, t, b ) . W e  ho ose the parametrization of ∂ ∞ X b y S 0 ∨ ∂ ∞ L su h that [ a, 0 , b ] = a , [ a, + ∞ , b ] = b , d ( π L ([ a, t, b ]) , p 0 ) = t , and the geo desi ra ys  π L ([ a, t, b ]) , [ a, t, b ]  are parallel transp orts of [ p 0 , a [ along the geo desi ra y [ p 0 , b [ , for 0 < t < + ∞ . F or ev ery t in ]0 , + ∞ [ and ev ery ( a, b ) in S 0 × ∂ ∞ L , let γ [ a,t,b ] b e the geo desi ra y or line starting from ξ 0 and ending at [ a, t, b ] . By the pro of of Lemma 3.4 and b y Lemma 3.2, w e ha v e − 2 c ′ 1 ( ǫ ) ≤ ℓ N ǫ L ( α ) − ftp L ( α ) ≤ 2 c ′ 1 ( ǫ ) + 2 ǫ, for ev ery α ∈ T 1 ξ 0 X . In partiular, if t = ftp L ( γ [ a,t,b ] ) > 2 c ′ 1 ( ǫ ) , then ℓ N ǫ L ( γ [ a,t,b ] ) > 0 , that is γ [ a,t,b ] meets the in terior of N ǫ L . By reduing to the ase X = H 3 R and L a geo desi line, it is easy to see that the map from [2 c ′ 1 ( ǫ ) , + ∞ [ to [0 , + ∞ [ dened b y t 7→ ℓ N ǫ L ( γ [ a,t,b ] ) is on tin uous and stritly inreasing, for ev ery xed ( a, b ) in S 0 × ∂ ∞ L . Hene, as ℓ N ǫ L ( γ [ a, 2 c ′ 1 ( ǫ ) ,b ] ) ≤ 4 c ′ 1 ( ǫ ) + 2 ǫ ≤ h < + ∞ , there exists a unique t a,b ∈ [2 c ′ 1 ( ǫ ) , + ∞ [ , dep ending on tin uously on ( a, b ) , su h that ℓ N ǫ L ( γ [ a,t a,b ,b ] ) = h . In partiular, the subset of p oin ts of ∂ ∞ X of the form [ a, t a,b , b ] for some ( a, b ) in S 0 × ∂ ∞ L is indeed a o dimension 1 top ologial submanifold of ∂ ∞ X , whi h is homeomorphi to the torus dim L − 1 × codim L − 1 . The statemen ts (b) and () follo w. If L has dimension 1 , and if γ ∈ T 1 ξ 0 X meets N ǫ L , then b y Lemma 2.5 and b y on v exit y , d ( π L ( γ ( −∞ )) , p 0 ) ≤ c ′ 1 ( ǫ ) . F or ev ery ξ in a omp onen t A 0 of A , if as ab o v e ξ = [ a, t a,b , b ] , then w e ha v e d ( π L ( ξ ) , p 0 ) = t a,b ≥ 2 c ′ 1 ( ǫ ) , hene A 0 separates γ ( − ∞ ) and b . This pro v es (a). Let us pro v e the last assertion of the lemma. Let κ = k f − ftp L k ∞ , and let A ′ b e a onneted omp onen t of A . F or ev ery u in L su h that d ( u, p 0 ) = h , let η 0 = [ a, h, b ] , on the same side of ∂ ∞ L as A ′ if co dim L = 1 , b e su h that π L ( η 0 ) = u . Let η t = [ a, h + t, b ] , whi h is on the same side of ∂ ∞ L as A ′ if co dim L = 1 . Note that f ( γ [ a,h + κ,b ] ) ≥ ftp L ( γ [ a,h + κ,b ] ) − κ = h, 36 and similarly , f ( γ [ a,h − κ,b ] ) ≤ h . By the in termediate v alue theorem, there exists t ∈ [ − κ, + κ ] su h that η t ∈ A ′ . Hene d ( u, π L ( η t )) = | t | ≤ κ .  No w w e pro eed with the pro of of the the remaining parts of Prop osition 3.7. Case (3) . By Lemma 3.8 , w e only ha v e to pro v e the seond laim, that there exists a geo desi ra y or line γ 0 starting from ξ 0 with γ 0 (+ ∞ ) b elonging to A 0 , su h that f ′ ( γ 0 ) ≤ h min 0 . Let κ = k f − ftp L k ∞ . Let p 0 (resp. p γ ) b e the p oin t of L the losest to ξ 0 (resp. γ (+ ∞ ) ), so that, in partiular, d ( p 0 , p γ ) = ftp L ( γ ) ≥ f ( γ ) − κ = h − κ > 0 . Let p ′ γ b e the p oin t on the geo desi line L 0 (on tained in L ) passing through p 0 and p γ on the opp osite side of p γ with resp et to p 0 , and at distane h from p 0 . By Lemma 3.9, there exists a geo desi line γ 0 starting from ξ 0 and ending at a p oin t in A 0 whose losest p oin t p γ 0 on L is at distane at most κ from p ′ γ . y ′ y p γ p 0 y 0 y ′ 0 γ 0 (+ ∞ ) ξ 0 γ (+ ∞ ) p γ 0 p ′ γ x 0 x γ 1 L 0 ⊂ L ≥ h − κ ≥ h − κ γ γ 0 Assume b y absurd that f ′ ( γ 0 ) > h min 0 . W e ha v e ℓ C ′ ( γ ) ≥ f ′ ( γ ) − κ ′ ≥ h ′ − κ ′ ≥ h min 0 − κ ′ > δ. Similarly ℓ C ′ ( γ 0 ) > δ , and, in partiular, γ 0 en ters C ′ . Let y ′ (resp. y ′ 0 ) b e the p oin t, p ossibly at innit y , where γ (resp. γ 0 ) exits C ′ . By Lemma 3.8 , γ 0 meets C b efore C ′ . Let x 0 (resp. y 0 ) b e the p oin t where γ 0 en ters in (resp. exits) C . As diam( C ∩ C ′ ) ≤ δ , w e ha v e y ′ 0 ∈ ] y 0 , γ 0 (+ ∞ )[ and y ′ ∈ ] y , γ (+ ∞ )[ , so that in partiular d ( y ′ 0 , L ) > ǫ and d ( y ′ , L ) > ǫ . Let γ 1 b e the geo desi line through y ′ and y ′ 0 . The p oin ts at innit y of γ 1 do not b elong to ∂ ∞ L 0 , so that ftp L 0 ( γ 1 ) and ℓ C ( γ 1 ) are nite. Note that b y strit on v exit y and b y Lemma 2.5 , w e ha v e d ( y ′ , [ p γ , γ (+ ∞ )[) < d ( y , [ p γ , γ (+ ∞ )[) ≤ c ′ 1 ( ǫ ) , and similarly d ( y ′ 0 , [ p γ 0 , γ 0 (+ ∞ )[) < c ′ 1 ( ǫ ) . Hene, with π L 0 the losest p oin t map to L 0 , whi h preserv es the b et w eenness and do es not inrease the distanes, ftp L 0 ( γ 1 ) ≥ d ( π L 0 ( y ′ 0 ) , π L 0 ( y ′ )) > d ( p γ 0 , p γ ) − 2 c ′ 1 ( ǫ ) = d ( p γ 0 , p 0 ) + d ( p 0 , p γ ) − 2 c ′ 1 ( ǫ ) ≥ h − κ + h − κ − 2 c ′ 1 ( ǫ ) = 2 h − 2 κ − 2 c ′ 1 ( ǫ ) . In partiular, b y Lemma 3.4 , ℓ C ( γ 1 ) ≥ ℓ N ǫ L 0 ( γ 1 ) ≥ ftp L 0 ( γ 1 ) − 2 c ′ 1 ( ǫ ) − 2 ǫ > 2 h − 2 κ − 4 c ′ 1 ( ǫ ) − 2 ǫ ≥ δ , 37 b y the denition of h min . Hene γ 1 meets C in a segmen t I of length > δ . But as y ′ 0 and y ′ are at a distane stritly bigger than ǫ of L , the segmen t I is on tained in [ y ′ , y ′ 0 ] , whi h is on tained in C ′ , b y on v exit y . This on tradits the assumption that diam( C ∩ C ′ ) ≤ δ . Case (4 ) . Let κ ′′ = k f ′ − ftp L ′ k ∞ . Note that f ′ ( γ ) ≥ h ′ ≥ h min 0 > δ + κ ′ . This is true under b oth assumptions on the v alue of h min 0 , as when h min 0 = 3 c ′ 1 ( ǫ ) + 3 ǫ + δ + κ ′′ , w e ha v e, b y Lemma 3.4 , δ + κ ′ ≤ δ + κ ′′ + 2 c ′ 1 ( ǫ ) + 2 ǫ < h min 0 . By Lemma 3.8 , w e only ha v e to pro v e the seond laim that there exists a geo desi line γ 0 starting from ξ 0 with γ 0 (+ ∞ ) b elonging to A 0 , su h that f ′ ( γ 0 ) ≤ h min 0 . W e rst onsider the ase C ′ = N ǫ L ′ where L ′ is a totally geo desi subspae of o di- mension at least 2 , with diam( C ∩ C ′ ) ≤ δ , and h ′ ≥ h min 0 = 3 c ′ 1 ( ǫ ) + 3 ǫ + δ + κ ′′ . Assume b y absurd that ev ery geo desi ra y or line α starting from ξ 0 with α ( ∞ ) ∈ A 0 meets C ′ with f ′ ( α ) > h min 0 . Let B ′ =  β ( ∞ ) : β ∈ T 1 ξ 0 X, ftp L ′ ( β ) > h min 0 − κ ′′  . By the absurdit y h yp othesis and the denition of κ ′′ , w e ha v e A 0 ⊂ B ′ . Let p ′ 0 b e the losest p oin t to ξ 0 on L ′ . Note that B ′ is a (top ologial) op en tubular neigh b ourho o d of ∂ ∞ L ′ , whose b er o v er a p oin t ξ in ∂ ∞ L ′ is the preimage of ρ ξ (] h min 0 − κ ′′ , + ∞ ]) b y the losest p oin t map from ∂ ∞ X to L ′ ∪ ∂ ∞ L ′ , where ρ ξ is the geo desi ra y with ρ (0) = p ′ 0 and ρ (+ ∞ ) = ξ . By Lemma 3.9 (a), let ξ 1 b e the p oin t at innit y of L separated from γ ( − ∞ ) b y A 0 . Note that as γ en ters C ′ at x ′ , and ξ 0 / ∈ C ′ , if p ′ γ ( −∞ ) is the losest p oin t to γ ( − ∞ ) on L ′ , then b y Lemma 2.5, w e ha v e d ( p ′ γ ( −∞ ) , p ′ 0 ) ≤ c ′ 1 ( ∞ ) < h min 0 − κ ′′ b y the denition of h min 0 . Hene the omplemen t of B ′ in ∂ ∞ X , whi h is onneted as co dim L ′ ≥ 2 , on tains γ ( − ∞ ) . As A 0 separates γ ( − ∞ ) from ξ 1 and is on tained in B ′ , it follo ws that B ′ on tains ξ 1 . x ′ 0 ξ 1 L ′ 0 ⊂ L ′ L p ′ ξ 1 p ′ u ξ 0 u p ′ 0 Let x ′ 0 b e the in tersetion p oin t of ] ξ 0 , p ′ 0 ] with ∂ C ′ . Lemma 2.5 implies that d ( x ′ , x ′ 0 ) ≤ c ′ 1 ( ǫ ) . Hene, b y on v exit y and as γ rst meets C and then C ′ , w e ha v e d  x, ] ξ 0 , p ′ 0 ]  ≤ c ′ 1 ( ǫ ) , whi h implies that there is a p oin t u in L at distane at most c ′ 1 ( ǫ ) + ǫ from ] ξ 0 , p ′ 0 ] . Let p ′ ξ 1 b e the losest p oin t to ξ 1 on L ′ , and L ′ 0 the geo desi line (on tained in L ′ ) through p ′ 0 and p ′ ξ 1 . As the losest p oin t map do es not inrease distanes, the losest p oin t p ′ u to u on L ′ 0 satises d ( p ′ 0 , p ′ u ) ≤ c ′ 1 ( ǫ ) + ǫ . Then, as the losest p oin t map to L ′ 0 preserv es the b et w eenness and as ξ 1 b elongs to B ′ , ftp L ′ 0 ( L ) ≥ d ( p ′ u , p ′ ξ 1 ) ≥ d ( p ′ ξ 1 , p ′ 0 ) − d ( p ′ 0 , p ′ u ) > h min 0 − κ ′′ − c ′ 1 ( ǫ ) − ǫ . 38 Therefore, using Lemma 3.4, diam( C ∩ C ′ ) ≥ diam( C ∩ N ǫ L ′ 0 ) ≥ ℓ N ǫ L ′ 0 ( L ) ≥ ftp L ′ 0 ( L ) − 2 c ′ 1 ( ǫ ) − 2 ǫ > h min 0 − κ ′′ − 3 c ′ 1 ( ǫ ) − 3 ǫ = δ , a on tradition. Assume no w that C ′ is an y ǫ -on v ex subset su h that C ∩ C ′ = ∅ , and that h ′ > h min 0 = κ ′′ . Let us pro v e that there exists a geo desi ra y or line γ 0 starting from ξ 0 with γ 0 (+ ∞ ) in A 0 , and a v oiding C ′ . This implies the result as in Case ( 1). By absurd, supp ose that for ev ery ξ in A 0 , the geo desi ra y or line γ ξ starting from ξ 0 and ending at ξ meets C ′ . By the rst laim (see Lemma 3.8), γ ξ meets the in terior of C b efore meeting C ′ . Let x ′ ξ b e the en tering p oin t of γ ξ in C ′ and y ξ b e its exiting p oin t out of C . As C and C ′ are disjoin t, note that ξ 0 , y ξ , x ′ ξ , ξ are in this order along γ ξ . The maps ξ 7→ y ξ and ξ 7→ x ′ ξ are injetiv e and on tin uous on a 0 (b y the strit on v exit y of C , as γ ξ meets the in terior of C ). W e kno w that A 0 is a top ologial sphere, b y Lemma 3.9 (a), separating the endp oin ts of L . Hene the subsets A 0 and S ′ = { x ′ ξ : ξ ∈ A 0 } are spheres, that are homotopi (b y the homotop y along γ ξ ) in the omplemen t of L in X ∪ ∂ ∞ X . By an homology argumen t, ev ery dis with b oundary S ′ in X ∪ ∂ ∞ X has to meet L . But b y on v exit y of C ′ , there exists a dis on tained in C ′ with b oundary S ′ (x a p oin t of S ′ and tak e the union of the geo desi ars from this p oin t to the other p oin ts of S ′ ). This on tradits the fat that C ∩ C ′ = ∅ .  Remarks. (1) In Case (2 ), w e ha v e h max ≥ h min if R is big enough, as c ′ 1 ( ǫ ) has a nite limit as ǫ → ∞ . (2) In Case (3) , if the o dimension of L is 1 , then w e ma y assume that γ meets L if γ meets L . Indeed, as w e ha v e seen in Lemma 3.9 (b), L ∪ ∂ ∞ L separates X ∪ ∂ ∞ X in to t w o onneted omp onen ts, and A (dened in the b eginning of the pro of ) has exatly t w o omp onen ts separated b y L ∪ ∂ ∞ L . If A + 0 is the omp onen t of A on the same side of ξ 0 from L ∪ ∂ ∞ L , and A − 0 the omp onen t of A on the other side, then a geo desi ra y or line starting from ξ 0 and ending in A + 0 do es not meet L (as L is totally geo desi), and an y geo desi line starting from ξ 0 and ending in A − 0 meets L , b y separation. This observ ation on the rossing prop ert y will b e used in the pro of of Corollary 5.12 to mak e sure that the lo ally geo desi ra y or line onstruted in the ourse of the pro of sta ys in the on v ex ore. (3) Case (4 ) is not true if C ′ is assumed to b e an y ǫ -on v ex subset, as sho wn b y taking X the real h yp erb oli 3 -spae, and C ′ the ǫ -neigh b orho o d of the (totally geo desi) h yp erb oli plane p erp endiular to L at a p oin t at distane h from the losest p oin t to ξ 0 on L : an y geo desi ra y or line α starting from ξ 0 , with ftp L ( α ) = h and meeting C ′ satises f ′ ( α ) = + ∞ for ev ery f ′ whi h is a κ ′ -p enetration map in C ′ . 4 The main onstrution 4.1 Unlouding the sky The aim of this setion is to pro v e the follo wing result, impro ving on our result in [PP1 ℄. The rst laim of Theorem 4.1 w as stated as Theorem 1.1. 39 Theorem 4.1 L et X b e a pr op er ge o desi CA T ( − 1) metri sp a e (having at le ast two p oints), with ar wise  onne te d b oundary ∂ ∞ X and extendible ge o desis. L et ( H α ) α ∈ A b e any family of b al ls or hor ob al ls with p airwise disjoint interiors. L et µ 0 = 1 . 534 . (1) F or every x in X − S α ∈ A H α , ther e exists a ge o desi r ay starting fr om x avoiding H α [ µ 0 ] for every α . (2) F or every α 0 in A , ther e exists a ge o desi line starting fr om the p oint at innity of H α 0 and avoiding H α [ µ 0 ] for every α 6 = α 0 . Remarks. (1) Note that b y its generalit y , Theorem 4.1 greatly impro v es the main results, Theorem 1.1 and Theorem 4.5, of [ PP1 ℄, where (exept for trees) X w as alw a ys assumed to b e a manifold, strit assumptions w ere made on the b oundary of X , and no denite v alue of µ 0 w as giv en exept in sp eial ases. But b esides this, an imp ortan t p oin t is that its pro of is a m u h simplied v ersion of the up oming main onstrution of Setion 4, and hene ould b e w elome as a guide for reading Setion 4.2. (2) Note that the onstan t µ 0 is not optimal, but not b y m u h. F or simpliial trees all of whose v erties ha v e degree at least 3 , the result is true, with an y µ 0 > 1 and this is optimal (though they do not satisfy the h yp otheses of the ab o v e result, the pro of is easy for them, see for instane [PP1 , Theo. 7.2 (3)℄). W e pro v ed in [ PP1 ℄ that the optimal v alue for the seond assertion of the theorem, when X = H n R , is µ 0 = − log (4 √ 2 − 5) ≈ 0 . 42 . Hene Theorem 4.1 (2) is not far from optimal, despite its generalit y . F urthermore, when X = H n R , a p ossible v alue of µ 0 for the rst assertion of the theorem that w as giv en in [PP1 , Theo. 7.1℄ w as log(2 + √ 5) − log(4 √ 2 − 5) ≈ 1 . 864 . Hene Theorem 4.1 (1) is ev en b etter than the orresp onding result in [PP1 ℄ when X = H n R , despite its generalit y . Pro of. W e start with the follo wing geometri lemma. F or ev ery µ ≥ 0 , dene ν ( µ ) = 2 e − µ 1 + √ 1 − e − 2 µ , (- 10 -) whi h is p ositiv e and dereasing from 2 to 0 as µ go es from 0 to + ∞ . Lemma 4.2 L et X b e a pr op er ge o desi CA T ( − 1) sp a e. L et H b e a b al l or hor ob al l in X and ξ 0 ∈ ( X ∪ ∂ ∞ X ) − ( H ∪ H [ ∞ ]) . L et µ ≥ log 2 b e at most the r adius of H , and let γ and γ ′ b e ge o desi r ays or lines starting at ξ 0 , me eting H [ µ ] , p ar ametrize d suh that γ ′ ( s ) , γ ( s ) ar e e quidistant to ξ 0 for some (hen e every) s , and that γ enters H at time 0 . (1) If x = γ (0) and x ′ ar e the p oints of entry in H of γ and γ ′ r esp e tively, then d ( x, x ′ ) ≤ ν ( µ ) . (2) F or every s ≥ 0 , we have d ( γ ( − s ) , γ ′ ( − s )) ≤ ν ( µ ) e − s . Pro of. Let ξ b e the en ter or p oin t at innit y of H , and let t, t ′ b e the en trane times of γ , γ ′ resp etiv ely in H [ µ ] . Note that t ≥ 0 as µ ≥ 0 . Let us pro v e rst that t ′ ≥ 0 to o. W e refer to Setion for the denition and prop erties of the map β ξ 0 , esp eially when ξ 0 ∈ X . Let u b e the p oin t on the geo desi ] ξ 0 , ξ [ su h that β ξ 0 ( x, u ) = 0 . By the on v exit y of the balls and horoballs, w e ha v e β ξ ( x, γ ′ (0)) ≤ β ξ ( x, u ) . Let us pro v e that β ξ ( x, u ) ≤ µ , whi h will hene imply that γ ′ en ters H [ µ ] at a non-negativ e time (whi h is t ′ ). 40 Glue the t w o omparison triangles ( ξ 0 , x, ξ ) and ( ξ , x, γ ( t ) ) in H 2 R for the geo desi triangles ( ξ 0 , x, ξ ) and ( ξ , x, γ ( t ) ) along their sides [ x, ξ ] . Let H b e the ball or horoball en tered at ξ su h that x ∈ ∂ H . By omparison, w e ha v e ∠ x ( ξ 0 , ξ ) ≤ π ≤ ∠ x ( ξ 0 , ξ ) + ∠ x ( ξ , γ ( t ) ) . Hene the geo desi ra y or line γ starting from ξ 0 and passing through x meets [ γ ( t ) , ξ ] , therefore it en ters H [ µ ] . Let u b e the p oin t on the geo desi [ ξ 0 , ξ ] su h that β ξ 0 ( x, u ) = 0 . As β ξ ( x, u ) = β ξ ( x, u ) , w e only ha v e to pro v e the result if X is the upp er halfspae mo del of the h yp erb oli plane H 2 R . W e ma y then assume that ξ 0 is the p oin t at innit y ∞ , and that H is the horoball with p oin t at innit y 0 and Eulidean diameter 1 (see the gure b elo w). But then, the v ertial o ordinate of γ (0) is at least 1 2 , and as e − µ ≤ 1 2 , the result follo ws: an y geo desi line starting from ξ 0 meets the horizon tal horosphere on taining γ (0) b efore p ossibly meeting H [ µ ] . ξ ξ 0 γ γ ( t ) x H H [ µ ] u No w, in order to pro v e b oth assertions of Lemma 4.2 , let us sho w that w e ma y assume that X = H 2 R . F or the rst one, glue the t w o omparison triangles ( ξ 0 , x, ξ ) and ( ξ 0 , x ′ , ξ ) for the geo desi triangles ( ξ 0 , x, ξ ) and ( ξ 0 , x ′ , ξ ) along their sides [ ξ 0 , ξ ] . As seen ab o v e, the geo desi lines γ (resp. γ ′ ) starting from ξ 0 and passing through x (resp. x ′ ) en ter H [ µ ] . And b y omparison, w e ha v e d ( x, x ′ ) ≤ d ( x, x ′ ) . F or the seond assertion, w e glue the t w o omparison triangles ( ξ 0 , γ ( t ) , γ ′ ( t ′ ) ) and ( ξ , γ ( t ) , γ ′ ( t ′ ) ) for the geo desi triangles ( ξ 0 , γ ( t ) , γ ′ ( t ′ )) and ( ξ , γ ( t ) , γ ′ ( t ′ )) along their isometri segmen ts [ γ ( t ) , γ ′ ( t ′ ) ] . As in the pro of of Lemma 2.5, the geo desi segmen t or ra y ] ξ 0 , γ ( t ) [ do es not meet the ball or horoball H [ µ ] en tered at ξ whose b oundary go es through γ ( t ) and γ ′ ( t ′ ) . By omparison, if H ′ is the ball or horoball en tered at ξ whose b oundary passes through the p oin t γ (0) on ] ξ 0 , γ ( t ) [ at distane t from γ ( t ) , then H ′ [ µ ] on tains H [ µ ] , so that ] ξ 0 , γ ( t ) ] and ] ξ 0 , γ ′ ( t ′ ) ] meet H ′ [ µ ] . F or ev ery s ≥ 0 , as t, t ′ ≥ 0 , if γ ( − s ) , γ ′ ( − s ) are the orresp onding p oin ts to γ ( − s ) , γ ′ ( − s ) on ] ξ 0 , γ ( t ) [ , ] ξ 0 , γ ′ ( t ′ ) [ resp etiv ely , then b y omparison d ( γ ( − s ) , γ ′ ( − s )) ≤ d ( γ ( − s ) , γ ′ ( − s ) ) . Hene w e ma y assume that X is the upp er halfspae mo del of the real h yp erb oli plane H 2 R . By homogene- it y and monotoniit y , it is suien t to pro v e the result for ξ 0 the p oin t at innit y ∞ , for H the horoball with p oin t at innit y 0 and Eulidean diameter 1 , and with γ and γ ′ dieren t and b oth tangen t to H [ µ ] . Then, b y an easy omputation, the Eulidean heigh t of the p oin t γ (0) is ν ′ ( µ ) = 1 2 (1 + √ 1 − e − 2 µ ) , so that the Eulidean heigh t of the p oin t γ ( − s ) is ν ′ ( µ ) e s . The h yp erb oli distane b et w een γ ( − s ) and γ ′ ( − s ) is hene at most e − µ ν ′ ( µ ) e s = ν ( µ ) e − s . With the ase s = 0 , this pro v es b oth assertions.  0 1 H γ (0 ) s e − µ 2 γ ′ ( − s ) γ ( − s ) 1 2 − e − µ 2 γ ′ (0) Pro of of Theorem 4.1 . Let X and ( H α ) α ∈ A b e as in the statemen t. Let ξ 0 b e either a p oin t in X − S α ∈ A H α or the p oin t at innit y of H α 0 for some α 0 in A . F or ev ery 41 µ 1 ≥ log 2 , dene the follo wing onstan ts, with ν the map in tro dued b efore Lemma 4.2, µ 2 = ν ( µ 1 ) > 0 , µ 3 = µ 1 + µ 2 > 0 , µ 4 = 2 µ 1 − 2 µ 2 . As µ 1 ≥ log 2 , ν is dereasing and ν (log 2) < log 2 , w e ha v e µ 4 > 0 . W e dene b y indution an initial segmen t N in N and the follo wing nite or innite sequenes • ( γ k ) k ∈ N of geo desi ra ys or lines starting from ξ 0 , • ( α k ) k ∈ N −{ 0 } of elemen ts in A , • ( t k ) k ∈ N of non-negativ e real n um b ers, • ( u k ) k ∈ N of maps u k : [0 , + ∞ [ → ]0 , + ∞ [ , su h that for ev ery k in N , the follo wing assertions hold: (1) If ξ 0 ∈ X , then γ k (0) = ξ 0 . Otherwise, γ k meets ∂ H α 0 at time 0 . (2) If k ≥ 1 , then γ k en ters H α k at the p oin t γ k ( t k ) and meets H α k [ µ 1 ] in one and only one p oin t. (3) If k ≥ 1 , then u k ( t ) = u k − 1 ( t ) + µ 2 e t − t k if t ≤ t k − 1 , and u k ( t ) = µ 3 if t > t k − 1 . (4) If k ≥ 1 , then t k ≥ µ 4 + t k − 1 . (5) If t ∈ [0 , t k [ , then the p oin t γ k ( t ) do es not b elong to S α ∈ A H α [ u k ( t )] . If ξ 0 ∈ X , let γ 0 b e a geo desi ra y starting from ξ 0 at time 0 . Otherwise, let γ 0 b e a geo desi line starting from ξ 0 and exiting H α 0 at time 0 . Su h a γ 0 exists b y the assumptions on X . Dene u 0 as the onstan t map t 7→ µ 3 . Let t 0 = 0 . The assertions (1)(5) are satised for k = 0 . Assume that γ k , t k , α k , u k are onstruted for 0 ≤ k ≤ n v erifying the assertions (1)(5). If γ n (] t n , + ∞ [) do es not en ter in the in terior of an y elemen t of the family ( H α [ µ 1 ]) α ∈ A , then dene N = [0 , n ] ∩ N , and the onstrution terminates. Otherwise, let H α n +1 [ µ 1 ] b e the rst elemen t of the family ( H α [ µ 1 ]) α ∈ A su h that the geo desi ra y γ n (] t n , + ∞ [) en ters in its in terior. Su h an elemen t exists as the H α 's ha v e disjoin t in teriors. Note that α n +1 6 = α n , as γ n do es not meet the in terior of H α n [ µ 1 ] b y (2). If ξ 0 ∈ X , let γ n +1 b e a geo desi ra y starting from ξ 0 at time 0 and meeting H α n +1 [ µ 1 ] in one and only one p oin t. This is p ossible as there exists a geo desi ra y starting from ξ 0 and a v oiding H α n +1 b y the prop erties of X (onsider for instane the extension to ] − ∞ , 0] of γ n ) and sine ∂ ∞ X is arwise onneted. If ξ 0 / ∈ X , let γ n +1 b e a geo desi line starting from ξ 0 , and meeting H α n +1 [ µ 1 ] in one and only one p oin t. Again, this is p ossible as ∂ ∞ X is arwise onneted. P arametrize γ n +1 su h that γ n +1 exits H α 0 at time 0 . In partiular, in b oth ases, the assertion (1) for k = n + 1 is satised. Dene t n +1 ≥ 0 su h that γ n +1 en ters H α n +1 at the p oin t γ n +1 ( t n +1 ) , so that the assertion (2) for k = n + 1 is satised. As γ n and γ n +1 b oth meet H α n +1 [ µ 1 ] and as µ 1 ≥ log 2 , it follo ws from Lemma 4.2 (2) that, for ev ery t ≤ t n +1 , d ( γ n +1 ( t ) , γ n ( t )) ≤ µ 2 e t − t n +1 . (- 11 -) 42 Dene τ n ≥ t n as the en trane time of γ n in H α n +1 . By Lemma 4.2 (1), as b oth γ n and γ n +1 meet H α n +1 [ µ 1 ] and µ 1 ≥ log 2 , w e ha v e d ( γ n +1 ( t n +1 ) , γ n ( τ n )) ≤ µ 2 . As H α n +1 and H α n ha v e disjoin t in teriors, and sine H α n and H α n [ µ 1 ] are at distane µ 1 , w e ha v e d ( γ n ( t n ) , γ n ( τ n )) ≥ 2 µ 1 . Hene d ( γ n ( t n ) , γ n ( t n +1 )) ≥ d ( γ n ( t n ) , γ n ( τ n )) − d ( γ n ( τ n ) , γ n +1 ( t n +1 )) − d ( γ n +1 ( t n +1 ) , γ n ( t n +1 )) ≥ 2 µ 1 − 2 µ 2 = µ 4 > 0 . Hene t n +1 − t n is p ositiv e and at least µ 4 , whi h pro v es the assertion (4) for k = n + 1 . Dene t 7→ u n +1 ( t ) b y the indution form ula in assertion (3). The only remaining assertion to v erify is (5). By absurd, assume that there exist some t in [0 , t n +1 [ and some α ∈ A su h that γ n +1 ( t ) b elongs to H α [ u n +1 ( t )] . As u n +1 ( t ) > 0 , the elemen t α is dieren t from α 0 if ξ 0 ∈ ∂ ∞ X , and it is also dieren t from α n +1 b y onstrution. By Equation (- 11 - ), the p oin t γ n ( t ) b elongs to H α [ u n +1 ( t ) − µ 2 e t − t n +1 ] . Assume rst that t > t n , so that u n +1 ( t ) = µ 3 . As µ 3 − µ 2 e t − t n +1 > µ 1 (w e annot ha v e t = t n +1 as H α and H α n +1 ha v e disjoin t in teriors), this implies that γ n ( t ) b elongs to the in terior of H α [ µ 1 ] . This on tradits the fat that H α n +1 [ µ 1 ] is the rst elemen t of the family ( H α [ µ 1 ]) α ∈ A enoun tered b y γ n (] t n , + ∞ [) in its in terior. Assume that t ≤ t n . Then γ n ( t ) b elongs to H α [ u n ( t )] . This on tradits the assertion (5) at step n . Th us, the assertions (1)(5) hold for all k ∈ N . Let us pro v e that the maps u n are uniformly b ounded from ab o v e b y µ 5 = µ 3 + µ 2 e µ 4 − 1 . As µ 4 > 0 , the sequene ( t k ) k ∈ N inreases to + ∞ . Fix t ≥ 0 . Let k = k ( t ) b e the unique non-negativ e in teger su h that t b elongs to ] t k − 1 , t k ] (b y on v en tion, t − 1 = −∞ ). Let us pro v e, b y indution on n , that u n ( t ) ≤ µ 3 + µ 2 n − k X j =1 e − µ 4 j . (Reall that an empt y sum is 0 ). This implies that u n ( t ) ≤ µ 5 . This is true if n = 0 , as u 0 ( t ) = µ 3 . Assume that the result is true for n . If t > t n , then u n +1 ( t ) = µ 3 , and the result is true. Otherwise, b y the prop ert y (3), w e ha v e u n +1 ( t ) = u n ( t ) + µ 2 e t − t n +1 . Note that t k − t n +1 ≤ − µ 4 ( n + 1 − k ) b y the prop ert y (4), and that t ≤ t k . Hene, b y indution, u n +1 ( t ) ≤ µ 3 + µ 2 n − k X j =1 e − µ 4 j + µ 2 e − µ 4 ( n +1 − k ) = µ 3 + µ 2 n +1 − k X j =1 e − µ 4 j . This pro v es the indution. Summarizing the ab o v e onstrution, there exist a sequene of geo desi ra ys or lines ( γ n ) n ∈ N starting from ξ 0 , and a sequene of times ( t n ) n ∈ N on v erging to + ∞ , su h that for ev ery t in [0 , t n ] , the p oin t γ n ( t ) do es not b elong to S α ∈ A H α [ µ 5 ] . (T ak e an ev en tually 43 onstan t sequene ( γ n ) n ∈ N if the onstrution stops at a nite stage, whi h is p ossible as µ 5 > µ 1 .) As ( t n ) n ∈ N gro ws at least linearly , the form ula ( - 11 - ) implies that ( γ n ( t )) n ∈ N is a Cau h y sequene, uniformly on ev ery ompat subset of non-negativ e t 's. Hene, the geo desi ra ys or lines γ n on v erge to a geo desi ra y or line a v oiding S α ∈ A H α [ µ 5 − ǫ ] , for ev ery ǫ > 0 . T aking µ 1 = 1 . 042 ≥ log 2 , w e an  he k that µ 5 < 1 . 5332 , hene the result follo ws.  Corollary 4.3 L et X and ( H α ) α ∈ A b e as in The or em 4.1. F or every x ∈ X , ther e exist t > 0 and a ge o desi r ay γ starting at x suh that γ ([ t, ∞ [) is  ontaine d in the  omplement of S α ∈ A H α [ µ 0 ] . Pro of. W e ma y assume that x ∈ H α 0 for some α 0 ∈ A , otherwise, Theorem 4.1 (1) applies (with t = 0 ). Let H ′ α = H α if α 6 = α 0 , and H ′ α 0 = H α 0 [ d ( x, ∂ H α 0 ) + 1] . Then x / ∈ X − S α ∈ A H ′ α . By Theorem 4.1 (1), let γ b e a geo desi ra y starting from x and a v oiding the H ′ α [ µ 0 ] 's. Let t = d ( x, ∂ H α 0 ) + 2 + 2 µ 0 + c ′ 1 ( ∞ ) . As k ℓ H α 0 − p h H α 0 k ∞ ≤ c ′ 1 ( ∞ ) b y Subsetion 3.1, the geo desi ra y γ ([ t, ∞ [) do es not meet H α 0 . The result follo ws.  Let e b e an end of a nite v olume omplete negativ ely urv ed Riemannian manifold V . Let h t e b e the Busemann funtion of e normalized to b e zero on the b oundary of the maximal Margulis neigh b ourho o d of e (see for instane [BK, HP3 , PP1 ℄, as w ell as the paragraph ab o v e Corollary 5.4). Our next result impro v es Theorem 7.4 (hene Corollary 1.2) in [PP1 ℄, with the same pro of as in [lo . it.℄, b y remo ving the te hnial assumptions on the manifold, and giving a univ ersal upp er b ound on h e ( V ) . Corollary 4.4 L et V b e a nite volume  omplete R iemannian manifold with dimension at le ast 2 and se tional urvatur e K ≤ − 1 . Then ther e exists a lose d ge o desi in V whose maximum height (with r esp e t to h t e ) is at most 1 . 534 .  4.2 The indutiv e onstrution Fix arbitrary onstan ts ǫ 0 ∈ R ∗ + ∪ {∞} and δ 0 , κ 0 ≥ 0 , and x an arbitrary p oin t ξ 0 in X ∪ ∂ ∞ X . Let ( C n ) n ∈ N b e a family of ǫ 0 -on v ex subsets of X su h that ξ 0 / ∈ C 0 ∪ ∂ ∞ C 0 , and let f 0 b e a κ 0 -p enetration map for C 0 . The aim of this setion is to onstrut b y indution a sequene of geo desi ra ys or lines in X , starting from ξ 0 and ha ving a suitable p enetration b eha viour in the C n 's. Presription of onstan ts. The follo wing onstan ts will app ear in the statemen t, or in the pro of, of the indutiv e onstrution: • c 1 = c ′ 1 ( ǫ 0 ) > 0 giv en b y Lemma 2.5 if ǫ 0 6 = ∞ and b y Lemma 2.11 if ǫ 0 = ∞ and ( f 0 , δ 0 ) 6 = ( ph C 0 , 0) ; otherwise c 1 = 1 19 ; • c 2 = c ′ 2 ( ǫ 0 ) > 0 giv en b y Equation (- 3 - ) if ǫ 0 6 = ∞ and b y (- 8 - ) otherwise; • c 3 = 2 sinh c 1 + c 2 e 2 c 1 sinh c 1 , whi h is p ositiv e, and dep ends on ǫ 0 ; • c 4 = c ′ 3 ( ǫ 0 ) sinh( c 1 + δ 0 ) + c 2 e − 3 c ′ 3 ( ǫ 0 ) s inh( c 1 + δ 0 ) − log 2 sinh c 1 , where c ′ 3 ( · ) is giv en b y Equation (- 5 - ) if ǫ 0 6 = ∞ and b y (- 9 - ) otherwise. Note that c 4 is p ositiv e, and dep ends on ǫ 0 , δ 0 ; 44 • c 5 = c 5 ( ǫ 0 , δ 0 ) = 2 max { c 2 , c ′ 3 ( ǫ 0 ) } sinh( c 1 + δ 0 ) , whi h is p ositiv e, and dep ends on ǫ 0 , δ 0 ; • c 6 = 3 c 4 + log 2 , whi h is p ositiv e, and dep ends on ǫ 0 , δ 0 ; • h 0 = h 0 ( ǫ 0 , δ 0 , κ 0 ) = max { δ 0 + κ 0 , c 0 ( ǫ 0 ) + κ 0 , h ′ ( ǫ 0 , sinh( δ 0 + c 1 )) , δ 0 + 2 c 1 + c 6 } , where c 0 ( · ) is giv en b y Equation (- 1 - ) if ǫ 6 = ∞ and b y (- 7 - ) otherwise, and h ′ ( · , · ) is giv en b y Equation (- 4 - ) if ǫ 0 6 = ∞ and b y (- 9 -) otherwise; • F or ev ery h ′ 0 ≥ 0 , let h ′ 1 = h ′ 1 ( ǫ 0 , δ 0 , h ′ 0 ) = h ′ 0 + 2 c 5 . Fix h ′ 0 ≥ h 0 and h ≥ h ′ 1 . Assumptions on the family ( C n ) n ∈ N . Assume that there exists at least one geo desi ra y or line γ 0 starting from ξ 0 and meeting C 0 with f 0 ( γ 0 ) = h , and that the follo wing onditions are satised. ( iii ) (Almost disjoin tness prop ert y) F or ev ery m, n in N with m 6 = n , the diameter of C n ∩ C m is at most δ 0 . ( iv ) (Lo al presription prop ert y) F or ev ery n in N − { 0 } su h that ξ 0 / ∈ C n ∪ ∂ ∞ C n , if there exists a geo desi ra y or line α starting from ξ 0 whi h meets rst C 0 and then C n with f 0 ( α ) = h and ℓ C n ( α ) ≥ h ′ 0 , then there exists a geo desi ra y or line α ′ , starting from ξ 0 whi h meets rst C 0 and then C n with f 0 ( α ′ ) = h and ℓ C n ( α ′ ) = h ′ 0 . Note that ( iii ) is satised with δ 0 = 0 if the C n 's ha v e disjoin t in terior. In Setion 5, w e will use Prop osition 3.7 to  he k ( iv ) for v arious appliations, with h ′ 0 = m ax { h 0 , h min 0 } and h ≥ m ax { h ′ 1 , h min } , for the v arious v alues of h min 0 , h min dened in the v arious ases of Prop osition 3.7. F or ev ery n in N su h that ξ 0 / ∈ C n ∪ ∂ ∞ C n , dene f n = ℓ C n : T 1 ξ 0 X → [0 , + ∞ ] , and for ev ery geo desi ra y or line γ starting from ξ 0 and meeting C n , let t − n ( γ ) , t + n ( γ ) ∈ ] − ∞ , + ∞ ] b e the en trane time and exit time of γ in and out of the on v ex subset C n resp etiv ely . The follo wing remark will b e used later on. Lemma 4.5 F or every n > 0 , for every ge o desi r ay or line γ starting fr om ξ 0 and entering C 0 at time t = 0 , suh that f 0 ( γ ) = h and γ (] δ 0 , + ∞ [) me ets C n , we have ξ 0 / ∈ C n ∪ ∂ ∞ C n and t − n ( γ ) > 0 . Pro of. Otherwise, as γ (] δ 0 , + ∞ [) meets C n and b y on v exit y , there exists ǫ > 0 su h that the geo desi segmen t γ ([0 , δ 0 + ǫ ]) is on tained in C n . By the P enetration prop ert y ( i ) of f 0 , the length of γ ∩ C 0 is at least h − κ 0 , whi h is bigger than δ 0 as h ≥ h ′ 1 > h ′ 0 ≥ h 0 ≥ δ 0 + κ 0 b y the denitions of h ′ 1 and h 0 . As γ en ters C 0 at time t = 0 , up to taking ǫ > 0 smaller, this implies that the geo desi segmen t γ ([0 , δ 0 + ǫ ]) is also on tained in C 0 . This on tradits the Almost disjoin tness prop ert y ( iii ) as n 6 = 0 .  Statemen t of the indutiv e onstrution. W e will dene b y indution an initial segmen t N in N , and nite or innite sequenes • ( γ k ) k ∈ N of geo desi ra ys or lines starting from ξ 0 , • ( n k ) k ∈ N of in tegers su h that ξ 0 / ∈ C n k ∪ ∂ ∞ C n k , 45 • ( u k ) k ∈ N of maps u k : [0 , + ∞ [ → [ h ′ 0 , h ′ 1 ] , su h that the follo wing assertions hold, for ev ery k in N , where w e use t ± k = t ± n k ( γ k ) to simplify notations. (1) The geo desi ra y or line γ k en ters C 0 at time t = 0 and f 0 ( γ k ) = h . (2) If k ≥ 1 , then γ k meets C n k with t − k ≥ 0 and f n k ( γ k ) = h ′ 0 . (3) If k ≥ 1 , then d ( γ k ( t ) , γ k − 1 ( t )) ≤ c 3 e t − t − k for ev ery t in [0 , t − k ] . (4) If k ≥ 1 , then u k ( t ) = sup s ∈ [0 , + ∞ [ : | s − t | ≤ c 4 e t − t − k u k − 1 ( s ) + c 5 e t − t − k for t ∈ [0 , t − k ] and u k ( t ) = h ′ 0 if t > t − k . (5) If k ≥ 1 , then t − k ≥ t − k − 1 + c 6 . (6) If k ≥ 1 , for ev ery n in N − { 0 } su h that γ k (] δ 0 , + ∞ [) meets C n with t − n ( γ k ) ≤ t − k , w e ha v e f n ( γ k ) ≤ u k ( t + n ( γ k ) − δ 0 ) . Note that b y Lemma 4.5 and b y (1), if γ k (] δ 0 , + ∞ [) meets C n , then ξ 0 / ∈ C n ∪ ∂ ∞ C n , so that, in partiular, t ± n ( γ k ) are w ell dened, and (6) do es mak e sense. Pro of of the indutiv e onstrution. By the assumptions, let γ 0 b e a geo desi ra y or line starting from ξ 0 and en tering C 0 at time t − 0 = 0 , su h that f 0 ( γ 0 ) = h . Let n 0 = 0 . Let u 0 : [0 , + ∞ [ → [ h ′ 0 , h ′ 1 ] b e the onstan t map with v alue h ′ 0 . As the onditions (2)(6) are empt y if k = 0 , the onstrution is done at step 0 . Let k ≥ 1 , and assume that γ 0 , n 0 , u 0 , . . . , γ k − 1 , n k − 1 , u k − 1 are onstruted. Note that u k − 1 ≥ h ′ 0 b y indution. If for ev ery n in N − { 0 } su h that γ k − 1 (] δ 0 , + ∞ [) meets C n , w e ha v e f n ( γ k − 1 ) ≤ u k − 1 ( t + n ( γ k − 1 ) − δ 0 ) , then w e stop and w e dene N = { 0 , 1 . . . , k − 1 } . Otherwise, let τ b e the greatest lo w er b ound of the t − n ( γ k − 1 ) 's tak en o v er all n in N − { 0 } su h that γ k − 1 (] δ 0 , + ∞ [) meets C n with f n ( γ k − 1 ) > u k − 1 ( t + n ( γ k − 1 ) − δ 0 ) . Let us pro v e that this lo w er b ound is in fat a minim um, attained for only one su h n . Let ǫ > 0 su h that h ′ 0 > δ 0 + ǫ , whi h is p ossible b y the denition of h 0 , as h ′ 0 ≥ h 0 . If t − n ( γ k − 1 ) and t − m ( γ k − 1 ) b elong to [ τ , τ + ǫ ] with f n ( γ k − 1 ) > u k − 1 ( t + n ( γ k − 1 ) − δ 0 ) and f m ( γ k − 1 ) > u k − 1 ( t + m ( γ k − 1 ) − δ 0 ) , assume for instane that t − n ( γ k − 1 ) ≤ t − m ( γ k − 1 ) . As f n = ℓ C n , f m = ℓ C m , u k − 1 ≥ h ′ 0 and t − m ( γ k − 1 ) − t − n ( γ k − 1 ) ≤ ǫ , the subsets C n and C m meet along a segmen t of length at least h ′ 0 − ǫ > δ 0 . By the Almost disjoin tness prop ert y (iii), this implies that n = m . In partiular, w e ha v e τ = t − n ( γ k − 1 ) for a unique n ∈ N − { 0 } , and w e denote this n b y n k ∈ N − { 0 } , so that γ k − 1 (] δ 0 , + ∞ [) meets C n k with f n k ( γ k − 1 ) > u k − 1 ( t + n k ( γ k − 1 ) − δ 0 ) ≥ h ′ 0 . (- 12 -) In partiular, ξ 0 / ∈ C n k ∪ ∂ ∞ C n k b y Lemma 4.5 and b y Assertion (1) at rank k − 1 . Note that n k 6 = n k − 1 , as f n k − 1 ( γ k − 1 ) = h ′ 0 b y the assertion (2) at rank k − 1 , whi h w ould on tradit Equation (- 12 - ) if n k = n k − 1 . 46 C 0 C n k γ k − 1 (0) γ k (0) γ k ( t − k ) γ k ( t + k ) γ k − 1 ( t + n k ( γ k − 1 )) γ k γ k − 1 γ k − 1 ( τ ) By Lemma 4.5, the geo desi ra y or line γ k − 1 rst en ters C 0 and then C n k . F urthermore, γ k − 1 satises (1) and f n k ( γ k − 1 ) ≥ h ′ 0 . Hene, b y the Lo al presription prop ert y ( iv ) , there exists a geo desi ra y or line γ k starting from ξ 0 that rst en ters C 0 and then C n k , with f 0 ( γ k ) = h and f n k ( γ k ) = h ′ 0 . Cho ose the parametrization in su h a w a y that γ k en ters C 0 at time 0 . In partiular, (1) and (2) hold for γ k , and t − k = t − n k ( γ k ) > 0 . Dene u k b y using the indution form ula giv en in the assertion (4). Before  he king (3)(6) for γ k , n k , u k , let us mak e t w o preliminary remarks. Lemma 4.6 W e have d ( γ k − 1 ( τ ) , γ k ( t − k )) ≤ c 1 and d ( γ k − 1 (0) , γ k (0)) ≤ c 1 . Pro of. By Lemma 2.5 if ǫ 0 6 = ∞ and Lemma 2.11 otherwise, w e ha v e d ( γ k − 1 ( τ ) , γ k ( t − k )) ≤ c ′ 1 ( ǫ 0 ) and d ( γ k − 1 (0) , γ k (0)) ≤ c ′ 1 ( ǫ 0 ) . By the denition of c 1 , w e hene only ha v e to pro v e Lemma 4.6 when ǫ 0 = ∞ , δ 0 = 0 and f 0 = ph C 0 . In this ase, as c 1 = 1 / 19 , c 2 = 5 / 2 , c 0 ( ∞ ) = 4 . 056 , κ 0 = 2 log (1 + √ 2) = c ′ 1 ( ∞ ) , c ′ 3 ( ∞ ) = 5 / 2 , easy omputations sho w that h 0 = h ′ ( ∞ , sinh c 1 ) = 3 sinh c 1 + c 0 ( ∞ ) + c ′ 1 ( ∞ ) ≈ 5 . 976 7 and, for future use, h ′ 1 ( ∞ , 0 , h 0 ( ∞ , 0 , c ′ 1 ( ∞ ))) ≈ 6 . 50 32 . (- 13 -) As ph C 0 ( γ k ) and ph C 0 ( γ k − 1 ) are equal to h ≥ h ′ 1 ≥ h ′ 0 ≥ h 0 , and sine h 0 / 2 ≥ log 2 , it follo ws from the denition of the map ph C 0 and from Lemma 4.2 (1) and (2) that d ( γ k − 1 (0) , γ k (0)) and similarly d ( γ k − 1 ( τ ) , γ k ( t − k )) are at most ν ( h 0 / 2) , where ν ( . ) is dened b y Equation (- 10 - ). An easy omputation sho ws that ν ( h 0 / 2) ≤ c 1 = 1 / 19 , whi h pro v es the result.  Lemma 4.7 W e have | τ − t − k | ≤ 2 c 1 . Pro of. Lemma 4.5, applied to n = n k and γ = γ k − 1 , implies that τ > 0 . W e ha v e seen that t − k > 0 . By the triangular inequalit y and the ab o v e lemma, w e ha v e | τ − t − k | ≤ 2 c 1 .  V eriation of (5). Note that τ = t − n k ( γ k − 1 ) > t − k − 1 . Otherwise, as t + n k ( γ k − 1 ) = τ + f n k ( γ k − 1 ) ≥ τ + h ′ 0 ≥ τ + h 0 > δ 0 b y Equation (- 12 - ) and b y the denition of h 0 , w e ha v e, b y the assertion (6) at step k − 1 , the inequalit y f n k ( γ k − 1 ) ≤ u k − 1 ( t + n k ( γ k − 1 ) − δ 0 ) , whi h on tradits the denition of n k , see Equation (- 12 - ). Let us rst pro v e that τ ≥ t − k − 1 + h 0 − δ 0 . Assume rst that τ ≥ t + k − 1 . Then, separating the ase k = 1 where f n k − 1 ( γ k − 1 ) = h ≥ h ′ 1 ≥ h ′ 0 from the ase k ≥ 2 where f n k − 1 ( γ k − 1 ) = h ′ 0 , w e ha v e τ − t − k − 1 ≥ t + k − 1 − t − k − 1 = f n k − 1 ( γ k − 1 ) ≥ h ′ 0 ≥ h 0 . (- 14 -) 47 Hene the result holds. Otherwise, t − k − 1 < τ < t + k − 1 . By on v exit y , γ k − 1 ( τ ) b elongs to C n k − 1 . Note that γ k − 1 ([ τ , τ + h 0 ]) is on tained in C n k , sine τ is the en trane time of γ k − 1 in C n k , and f n k ( γ k − 1 ) ≥ h ′ 0 ≥ h 0 . If τ + δ 0 < t + k − 1 , as ℓ C n k ( γ k − 1 ) ≥ h 0 > δ 0 b y the denition of h 0 , then C n k ∩ C n k − 1 on tains a geo desi segmen t of length bigger than δ 0 . This on tradits the Almost disjoin tness prop ert y ( iii ) sine n k 6 = n k − 1 . Hene τ ≥ t + k − 1 − δ 0 ≥ t − k − 1 + h 0 − δ 0 b y Equation (- 14 - ), and the result holds. No w, b y Lemma 4.7 , t − k − t − k − 1 ≥ τ − 2 c 1 − t − k − 1 ≥ h 0 − δ 0 − 2 c 1 ≥ c 6 b y the denition of h 0 . Therefore, the assertion (5) holds at rank k . V eriation of (4). W e only ha v e to  he k that u k has v alues in [ h ′ 0 , h ′ 1 ] . W e start b y pro ving the follo wing easy but tedious general lemma. Lemma 4.8 L et c, c ′ , c ′′ , h ∗ ≥ 0 , let M b e an initial se gment in N , let ( t n ) n ∈ M b e a se quen e of non-ne gative r e al numb ers, and let ( u n : [0 , + ∞ [ → [0 , + ∞ [) n ∈ M b e a se quen e of maps. Assume that u 0 has  onstant value h ∗ , and that for every n in M − { 0 } , we have t n − t n − 1 ≥ c ′′ , u n ( t ) = h ∗ if t > t n and if t ≤ t n , then u n ( t ) = c e t − t n + sup s ∈ [0 , + ∞ [ : | s − t | ≤ c ′ e t − t n u n − 1 ( s ) . If c ′′ ≥ 3 c ′ + log 2 , then for every t ∈ [0 , + ∞ [ , for every n in M , we have h ∗ ≤ u n ( t ) ≤ h ∗ + 2 c . T o pro v e that u k has v alues in [ h ′ 0 , h ′ 1 ] , w e apply Lemma 4.8 with c = c 5 , c ′ = c 4 , c ′′ = c 6 , h ∗ = h ′ 0 , M = { 0 , 1 , . . . , k } and ( t i ) i ∈ M = ( t − i ) 1 ≤ i ≤ k . Its h yp otheses are satised b y the denition of the onstan t c 6 , b y the assertion (5) at rank less than or equal to k , that w e just pro v ed, and b y the denition of u k and the assertion (4) for u i with 0 ≤ 1 ≤ k − 1 . Hene the map u k do es ha v e v alues in [ h ′ 0 , h ′ 1 ] , b y the denition of h ′ 1 . Pro of of Lemma 4.8. First note that b y an easy indution, whatev er the v alue of c ′′ is, for ev ery t ∈ [0 , + ∞ [ and n ∈ M , w e ha v e u n ( t ) ≥ h ∗ . Let c ′′ ≥ 3 c ′ + log 2 , t ∈ [0 , + ∞ [ and n ∈ M . Let us pro v e that u n ( t ) ≤ h ∗ + 2 c . W e ma y assume that t ≤ t n and that n ≥ 1 . Dene t − 1 = − 2 c ′ − 1 . Let m b e the unique elemen t in N su h that t m − 1 + 2 c ′ < t ≤ t m + 2 c ′ . Let N = n − m ≥ 0 . Note that for ev ery in teger k with 0 ≤ k ≤ N , w e ha v e t n − k − t m ≥ ( n − m − k ) c ′′ hene t − t n − k ≤ 2 c ′ − ( N − k ) c ′′ . (- 15 -) Consider the nite sequene ( x k ) 0 ≤ k ≤ N dened b y x 0 = 0 and x k +1 = x k + e c ′ x k − ( N − k ) c ′′ +2 c ′ for 0 ≤ k ≤ N − 1 . Let us pro v e b y indution on k that x k ≤ e − ( N − k ) c ′′ , whi h in partiular implies that x N ≤ 1 . (- 16 -) 48 Indeed, the result is true for k = 0 . Assume it to b e true for some k ≤ N − 1 . Then x k +1 ≤ e − ( N − k ) c ′′ + e c ′ e − ( N − k ) c ′′ − ( N − k ) c ′′ +2 c ′ ≤ e − ( N − k − 1) c ′′  e − c ′′ + e − c ′′ +3 c ′  ≤ e − ( N − k − 1) c ′′ as c ′′ ≥ 3 c ′ + log 2 ≥ log (1 + e 3 c ′ ) . Let us no w pro v e b y indution on k that, for 0 ≤ k ≤ N , w e ha v e u n ( t ) ≤ sup | s − t | ≤ c ′ x k u n − k ( s ) + c x k . (- 17 -) This is true if k = 0 , assume it is true for some k ≤ N − 1 . In partiular, n − k ≥ 1 . F or ev ery s ∈ [0 , + ∞ [ su h that | s − t | ≤ c ′ x k , w e ha v e u n − k ( s ) ≤ sup | s ′ − s | ≤ c ′ e s − t n − k u n − k − 1 ( s ′ ) + c e s − t n − k (this is true b y denition if s ≤ t n − k , and also true otherwise as then u n − k ( s ) = h ∗ and u n − k − 1 ( s ′ ) ≥ h ∗ for ev ery s ′ ). Hene b y the triangular inequalit y and the equation (- 15 - ), u n ( t ) ≤ sup | s ′ − t | ≤ c ′ x k + c ′ e t + c ′ x k − t n − k u n − k − 1 ( s ′ ) + c x k + ce t + c ′ x k − t n − k ≤ sup | s ′ − t | ≤ c ′ x k + c ′ e c ′ x k +2 c ′ − ( N − k ) c ′′ u n − k − 1 ( s ′ ) + c x k + ce c ′ x k +2 c ′ − ( N − k ) c ′′ = sup | s ′ − t | ≤ c ′ x k +1 u n − k − 1 ( s ′ ) + c x k +1 , whi h pro v es the indutiv e form ula ( - 17 - ). Finally , let us pro v e that u n ( t ) ≤ h ∗ + 2 c , whi h nishes the pro of of the lemma. T ak e k = N in the indutiv e form ula (- 17 - ) , and note that n − N = m . F or ev ery ǫ > 0 , let s ∈ [0 , + ∞ [ with | s − t | ≤ c ′ x N su h that sup | s ′ − t | ≤ c ′ x N u m ( s ′ ) ≤ u m ( s ) + ǫ . If s > t m or m = 0 , then u m ( s ) = h ∗ , hene b y the inequalit y (- 16 - ), u n ( t ) ≤ sup | s ′ − t | ≤ c ′ x N u m ( s ′ ) + c x N ≤ h ∗ + ǫ + c , and the result holds. Otherwise, s ≤ t m and m ≥ 1 . F or ev ery s ′ ∈ [0 , + ∞ [ su h that | s ′ − s | ≤ c ′ e s − t m , w e ha v e s ′ ≥ s − c ′ ≥ t − 2 c ′ > t m − 1 . Again, the denition of s and the inequalit y (- 16 - ) giv es u n ( t ) ≤ u m ( s ) + ǫ + c x N = sup | s ′ − s | ≤ c ′ e s − t m u m − 1 ( s ′ ) + c e s − t m + ǫ + c x N ≤ ǫ + h ∗ + 2 c , and the result also holds.  V eriation of (3). Let t b e in [0 , t − k ] . Reall that d ( γ k − 1 ( τ ) , γ k ( t − k )) ≤ c 1 , hene w e ha v e d ( γ k − 1 ( τ ) , γ k ) ≤ c 1 b y Lemma 4.6. By Lemma 2.1 , w e ha v e d ( γ k − 1 (0) , γ k ) ≤ e − τ sinh c 1 . By the P enetration prop ert y ( i ) of f 0 and the denition of h 0 , w e ha v e ℓ C 0 ( γ k ) ≥ f 0 ( γ k ) − κ 0 = h − κ 0 ≥ h 0 − κ 0 ≥ c 0 ( ǫ 0 ) . 49 Th us, b y Lemma 2.7 if ǫ 0 6 = ∞ and b y Lemma 2.13 if ǫ 0 = ∞ , and b y the denition of c 2 , w e ha v e d ( γ k − 1 (0) , γ k (0)) ≤ c 2 e − τ sinh c 1 . (- 18 -) W e refer to Setion 2.1 for the denition and prop erties of the map β ξ 0 . It follo ws from the inequalit y (- 18 - ) that | β ξ 0 ( γ k − 1 ( t ) , γ k ( t )) | = | β ξ 0 ( γ k − 1 (0) , γ k (0)) | ≤ d ( γ k − 1 (0) , γ k (0)) ≤ c 2 e − τ sinh c 1 . (- 19 -) F or ev ery s in R , let γ k − 1 ( s ′ ) b e the p oin t on the geo desi line γ k − 1 su h that the equalit y β ξ 0 ( γ k − 1 ( s ′ ) , γ k ( s )) = 0 holds. F or ev ery p oin t p ∈ γ k − 1 , w e ha v e d ( p, γ k − 1 ( t ′ )) =   β ξ 0 ( p, γ k − 1 ( t ′ ))   =   β ξ 0 ( p, γ k ( t ))   ≤ d ( p, γ k ( t )) , (- 20 -) Using the triangle inequalit y with the p oin t p the losest to γ k ( t ) on γ k − 1 , Lemma 2.1 and Lemma 4.6, w e hene ha v e the follo wing inequalit y d ( γ k ( t ) , γ k − 1 ( t ′ )) ≤ 2 d ( γ k ( t ) , γ k − 1 ) ≤ 2 e t − t − k sinh d ( γ k ( t − k ) , γ k − 1 ( τ )) ≤ 2 e t − t − k sinh c 1 . (- 21 -) Note that, using Equation (- 20 - ) with p = γ k − 1 ( t ) and the inequalities (- 19 - ), d ( γ k − 1 ( t ) , γ k − 1 ( t ′ )) = | β ξ 0 ( γ k − 1 ( t ) , γ k ( t )) | ≤ c 2 e − τ sinh c 1 .. Hene, b y the inequalit y (- 21 - ), w e ha v e d ( γ k ( t ) , γ k − 1 ( t )) ≤ d ( γ k ( t ) , γ k − 1 ( t ′ )) + d ( γ k − 1 ( t ′ ) , γ k − 1 ( t )) ≤ 2 e t − t − k sinh c 1 + c 2 e − τ sinh c 1 . As τ ≥ t − k − 2 c 1 b y Lemma 4.7 , and b y the denition of c 3 , w e get d ( γ k ( t ) , γ k − 1 ( t )) ≤ c 3 e t − t − k , whi h pro v es the assertion (3) at rank k . V eriation of (6). By absurd, assume that there exists n ∈ N − { 0 } su h that γ k (] δ 0 , + ∞ [) meets C n (so that in partiular ξ 0 / ∈ C n ∪ ∂ ∞ C n b y Lemma 4.5), with t − n ( γ k ) ≤ t − k and f n ( γ k ) > u k ( t + n ( γ k ) − δ 0 ) . (- 22 -) T o simplify notation, let s ± k = t ± n ( γ k ) , x = γ k ( s − k ) , y = γ k ( s + k ) , and, as w e will pro v e later on that γ k − 1 also meets C n , let s ± k − 1 = t ± n ( γ k − 1 ) , x ′ = γ k − 1 ( s − k − 1 ) , y ′ = γ k − 1 ( s + k − 1 ) . γ k ( t − k ) γ k ( t + k ) γ k − 1 ( t + n k ( γ k − 1 )) γ k γ k − 1 C n C n k γ k − 1 ( τ ) y = γ k ( s + k ) x = γ k ( s − k ) x ′ = γ k − 1 ( s − k − 1 ) y ′ = γ k − 1 ( s + k − 1 ) 50 Note that s + k ≤ t − k + δ 0 . Otherwise, as s − k ≤ t − k and b y on v exit y , there exists ǫ > 0 su h that γ k ([ t − k , t − k + δ 0 + ǫ ]) is on tained in C n . As t + k − t − k = h ′ 0 ≥ h 0 > δ 0 , up to making ǫ smaller, the geo desi segmen t γ k ([ t − k , t − k + δ 0 + ǫ ]) is also on tained in C n k . Hene n is equal to n k b y the Almost disjoin tness prop ert y (iii). But f n k ( γ k ) = h ′ 0 and, b y Equation (- 22 - ), w e ha v e f n ( γ k ) > u k ( s + k − δ 0 ) ≥ h ′ 0 , so that n annot b e equal to n k . By Lemma 2.1 applied to the geo desi triangle with v erties γ k ( t − k + δ 0 ) , γ k − 1 ( τ ) , ξ 0 , and as d ( γ k ( t − k ) , γ k − 1 ( τ )) ≤ c 1 b y Lemma 4.6, w e ha v e d ( y , γ k − 1 ) ≤ e − d ( γ k ( t − k + δ 0 ) ,y ) sinh d ( γ k ( t − k + δ 0 ) , γ k − 1 ( τ )) ≤ e s + k − t − k − δ 0 sinh( δ 0 + c 1 ) (- 23 -) whi h is, in partiular, at most sinh( δ 0 + c 1 ) . Let q ′ b e the losest p oin t to y on γ k − 1 , and let p (resp. q ) b e the losest p oin t to x ′ (resp. q ′ ) on γ k . Then d ( x ′ , p ) ≤ d ( x, x ′ ) ≤ c ′ 1 ( ǫ 0 ) b y Lemma 2.5 if ǫ 0 6 = ∞ and Lemma 2.11 otherwise. As losest p oin t maps do not inrease distanes, w e ha v e d ( y , q ) ≤ d ( y , q ′ ) = d ( y , γ k − 1 ) ≤ sinh( δ 0 + c 1 ) . Note that d ( x, y ) = f n ( γ k ) > h ′ 0 ≥ h 0 ≥ h ′ ( ǫ 0 , sinh( δ 0 + c 1 )) ≥ sinh( δ 0 + c 1 ) + c ′ 1 ( ǫ 0 ) , (- 24 -) b y the denition of h 0 and of h ′ ( · , · ) in Equation (- 4 - ) if ǫ 0 6 = ∞ and b y (- 9 - ) otherwise. Similarly , w e ha v e d ( x, y ) ≥ h 0 ≥ c 0 ( ǫ 0 ) . Hene ξ 0 , x ′ , q ′ are in this order on γ k . Therefore, b y on v exit y , d ( x ′ , γ k ) ≤ d ( q ′ , γ k ) ≤ d ( q ′ , y ) = d ( y , γ k − 1 ) . Hene, b y Lemma 2.7 if ǫ 0 6 = ∞ and Lemma 2.13 otherwise, and b y the inequalit y (- 23 - ), w e ha v e d ( x, x ′ ) ≤ c 2 d ( x ′ , γ k ) ≤ c 2 e s + k − t − k − δ 0 sinh( δ 0 + c 1 ) . (- 25 -) F urthermore, as w e ha v e seen that d ( x, y ) ≥ h ′ ( ǫ 0 , sinh( δ 0 + c 1 )) and b y the inequalit y (- 23 - ), it follo ws from Lemma 2.8 if ǫ 0 6 = ∞ and b y Lemma 2.14 otherwise, that the geo desi line γ k − 1 meets C n and one of the follo wing t w o assertions hold : d ( y , y ′ ) ≤ c ′ 3 ( ǫ 0 ) d ( x ′ , γ k ) ≤ c ′ 3 ( ǫ 0 ) e s + k − t − k − δ 0 sinh( δ 0 + c 1 ) (- 26 -) or d ( x ′ , y ′ ) ≥ d ( x, y ) . (- 27 -) Before obtaining a on tradition from b oth of these assertions, w e pro v e a te hnial result. Lemma 4.9 W e have δ 0 < s − k − 1 < τ , so that γ k − 1 (] δ 0 , + ∞ [) me ets C n with t − n ( γ k − 1 ) < τ . Pro of. Assume rst b y absurd that s − k − 1 ≤ δ 0 . If s − k − 1 ∈ ]0 , δ 0 ] , w e ha v e b y the triangular inequalit y , Lemma 4.6 and the inequalit y (- 25 - ) , s − k = d ( γ k (0) , γ k ( s − k )) ≤ d ( γ k (0) , γ k − 1 (0)) + d ( γ k − 1 (0) , γ k − 1 ( s − k − 1 )) + d ( γ k − 1 ( s − k − 1 ) , γ k ( s − k )) ≤ c 1 + δ 0 + c 2 sinh( δ 0 + c 1 ) . 51 Let z 0 and z s − k − 1 b e the losest p oin ts on γ k to γ k − 1 (0) and γ k − 1 ( s − k − 1 ) , resp etiv ely . If s − k − 1 ≤ 0 , then as the losest p oin t pro jetion do es not inrease distanes, w e ha v e s − k = d ( γ k (0) , γ k ( s − k )) ≤ d  γ k (0) , z 0 )  + d  z 0 , γ k ( s − k )  ≤ d  γ k (0) , z 0 )  + d  z s − k − 1 , γ k ( s − k )  ≤ d ( γ k (0) , γ k − 1 (0)) + d ( γ k − 1 ( s − k − 1 ) , γ k ( s − k )) ≤ c 1 + c 2 sinh( δ 0 + c 1 ) . Hene, b y the denition of c 5 and as c ′ 3 ( ǫ 0 ) ≥ 1 (see the equation (- 5 - ) if ǫ 0 6 = ∞ or (- 9 - ) otherwise), w e ha v e c 5 ≥ c 2 sinh( δ 0 + c 1 ) + c ′ 3 ( ǫ 0 ) sinh( δ 0 + c 1 ) ≥ c 2 sinh( δ 0 + c 1 ) + δ 0 + c 1 ≥ s − k . No w ℓ C n ( γ k ) ≥ h 0 > δ 0 b y the denition of h 0 , and ℓ C 0 ( γ k ) ≥ f 0 ( γ k ) − κ 0 = h − κ 0 ≥ h ′ 1 − κ 0 ≥ h 0 + 2 c 5 − κ 0 > δ 0 + c 5 ≥ δ 0 + s − k . As s − k ≥ 0 is the en trane time of γ k in C n , this implies that diam( C 0 ∩ C n ) > δ 0 . As n 6 = 0 , this on tradits the Almost disjoin tness prop ert y ( iii ) , hene δ 0 < s − k − 1 . Assume no w b y absurd that s − k − 1 ≥ τ . Then as in the ase s − k − 1 ≤ 0 , w e get t − k − s − k ≤ d ( γ k − 1 ( τ ) , γ k ( t − k )) + d ( γ k − 1 ( s − k − 1 ) , γ k ( s − k )) ≤ c 1 + c ′ 1 ( ǫ 0 ) , b y Lemma 4.6 , and b y Lemma 2.5 if ǫ 0 6 = ∞ and Lemma 2.11 otherwise. W e ha v e seen in the inequalities (- 24 - ) that h 0 ≥ sinh( δ 0 + c 1 ) + c ′ 1 ( ǫ 0 ) > δ 0 + c 1 + c ′ 1 ( ǫ 0 ) . Hene t − k ≥ s + k − δ 0 ≥ s − k + h 0 − δ 0 > s − k + c 1 + c ′ 1 ( ǫ 0 ) , a on tradition. Hene s − k − 1 < τ .  Assume rst that the inequalit y (- 26 - ) holds. As s − k ≥ 0 b y Lemma 4.5 and b y the denition of h 0 , w e ha v e s + k > h ′ 0 + s − k ≥ h 0 ≥ δ 0 + 2 c 1 + c 6 . Hene, as τ ≥ t − k − 2 c 1 b y Lemma 4.7 , w e ha v e e − τ ≤ e − c 6 e s + k − δ 0 − t − k . By the denition of c 6 and of c 4 , w e ha v e c 6 = 3 c 4 + log 2 ≥ 3 c ′ 3 ( ǫ 0 ) sinh( c 1 + δ 0 ) + l og 2 . By the triangular inequalit y sine s + k − 1 ≥ 0 b y 4.9, b y the equations (- 26 - ) and (- 18 - ), and b y the denition of c 4 , w e hene ha v e | s + k − s + k − 1 | ≤ d ( y, y ′ ) + d ( γ k (0) , γ k − 1 (0)) ≤ c ′ 3 ( ǫ 0 ) e s + k − t − k − δ 0 sinh( δ 0 + c 1 ) + c 2 e − τ sinh c 1 ≤ c 4 e s + k − δ 0 − t − k . (- 28 -) 52 By the Lips hitz prop ert y ( ii ) of f n = ℓ C n (as n 6 = 0 ), b y the inequalities (- 25 - ) and (- 26 - ), and b y the denition of c 5 , w e ha v e | f n ( γ k − 1 ) − f n ( γ k ) | ≤ 2 max { d ( x, x ′ ) , d ( y , y ′ ) } ≤ 2 max { c 2 e s + k − δ 0 − t − k sinh( δ 0 + c 1 ) , c ′ 3 ( ǫ 0 ) e s + k − δ 0 − t − k sinh( δ 0 + c 1 ) } ≤ c 5 e s + k − δ 0 − t − k . (- 29 -) By the inequalities (- 22 - ) and (- 29 - ), b y the minimalit y prop ert y of τ , and b y Lemma 4.9 , w e ha v e u k ( s + k − δ 0 ) < f n ( γ k ) ≤ f n ( γ k − 1 ) + c 5 e s + k − δ 0 − t − k ≤ u k − 1 ( s + k − 1 − δ 0 ) + c 5 e s + k − δ 0 − t − k . Assume no w that the inequalit y (- 27 - ) holds instead of the inequalit y (- 26 - ). Then f n ( γ k ) ≤ f n ( γ k − 1 ) , so w e again ha v e that u k ( s + k − δ 0 ) < u k − 1 ( s + k − 1 − δ 0 ) + c 5 e s + k − δ 0 − t − k . As | ( s + k − 1 − δ 0 ) − ( s + k − δ 0 ) | ≤ c 4 e s + k − δ 0 − t − k b y the inequalit y (- 28 - ), this on tradits the assertion (4) on the map u k . Hene the assertion (6) at rank k is v eried. The main orollary of the onstrution. The ab o v e indutiv e onstrution will only b e used in this pap er through the follo wing summarizing statemen t. Prop osition 4.10 L et X b e a pr op er ge o desi CA T( − 1) metri sp a e. L et ǫ 0 in R ∗ + ∪ {∞} , δ 0 , κ 0 ≥ 0 and ξ 0 ∈ X ∪ ∂ ∞ X . L et h ′ 0 ≥ h 0 ( ǫ 0 , δ 0 , κ 0 ) and h ≥ h ′ 1 = h ′ 1 ( ǫ 0 , δ 0 , h ′ 0 ) . L et ( C n ) n ∈ N b e a  ol le tion of ǫ 0 - onvex subsets of X whih satises the assertions ( iii ) and ( iv ) , and with ξ 0 / ∈ C 0 ∪ ∂ ∞ C 0 . L et f 0 : T 1 ξ 0 X → [0 , + ∞ ] b e a  ontinuous κ 0 - p enetr ation map in C 0 . Assume that ther e exists a ge o desi r ay or line γ 0 starting fr om ξ 0 with f 0 ( γ 0 ) = h . Then ther e exists a ge o desi r ay or line γ ∞ starting fr om ξ 0 , entering C 0 at time t = 0 with f 0 ( γ ∞ ) = h , suh that ℓ C n ( γ ∞ ) ≤ h ′ 1 for every n in N − { 0 } suh that γ ∞ (] δ 0 , + ∞ [) me ets C n . Pro of. Apply the main onstrution of the previous subsetions with initial input a geo desi ra y or line γ 0 en tering C 0 at time t = 0 with f 0 ( γ 0 ) = h , to get nite or innite sequenes ( γ k ) k ∈ N , ( n k ) k ∈ N , ( u k ) k ∈ N satisfying the assertions (1)(6). If N is nite, with maxim um N , dene γ k = γ N for k > N . Then the sequene ( γ k ) k ∈ N on v erges to a geo desi ra y or line γ ∞ = γ N in T 1 ξ 0 X . If N is innite, as X is omplete, it follo ws from the assertions (3) and (5), b y an easy geometri series argumen t, that the sequene ( γ k ) k ∈ N on v erges in T 1 ξ 0 X to a geo desi ra y or line γ ∞ starting from ξ 0 and en tering C 0 at time t = 0 , as C 0 is losed and on v ex. By the on tin uit y of f 0 and the assertion (1), w e ha v e f 0 ( γ ∞ ) = h . Supp ose b y absurd that there exists n in N − { 0 } su h that γ ∞ (] δ 0 , + ∞ [) meets C n and ℓ C n ( γ ∞ ) > h ′ 1 > 0 . In partiular, γ ∞ (] δ 0 , + ∞ [) meets the in terior of C n and ξ 0 / ∈ C n ∪ ∂ ∞ C n b y Lemma 4.5 . F urthermore, it follo ws from the denition of the stopping time, and the fat that u k ≤ h ′ 1 for ev ery k , that N is innite. Hene, as the γ k 's on v erge to γ ∞ , and b y the on tin uit y of ℓ C n , if k is big enough, then γ k (] δ 0 , + ∞ [) meets C n and ℓ C n ( γ k ) > h ′ 1 . In partiular, t + n ( γ k ) > δ 0 . Note that t − n ( γ k ) , whi h is at distane at most c ′ 1 ( ǫ 0 ) from t − n ( γ ∞ ) b y Lemma 2.5 if ǫ 0 6 = ∞ and Lemma 2.11 otherwise, is b ounded as k tends to 53 ∞ . Hene if k is big enough, then t − n ( γ k ) is less than t − k , as t − k on v erges to + ∞ when k → + ∞ b y the assertion (5). This on tradits the assertion (6), as u k ≤ h ′ 1 .  Remark. If X, ǫ 0 , δ 0 , κ 0 , ξ 0 , h ′ 0 , h, ( C n ) n ∈ N , f 0 satisfy the h yp otheses in the statemen t of Prop osition 4.10, and if for ev ery n su h that ξ 0 / ∈ C n ∪ ∂ ∞ C n , w e ha v e a κ -p enetration map g n : T 1 ξ 0 X → [0 , + ∞ ] for some onstan t κ ≥ 0 , then Prop osition 4.10 implies that there exists a geo desi ra y or line γ ∞ starting from ξ 0 , en tering C 0 at time t = 0 with f 0 ( γ ∞ ) = h , su h that g n ( γ ∞ ) ≤ h ′ 1 + κ for ev ery n in N − { 0 } su h that γ ∞ (] δ 0 , + ∞ [) meets C n . W e will apply this observ ation to more general p enetration maps than the ℓ C n 's, in Setion 5. The next orollary yields geo desi lines with the presrib ed p enetration in C 0 , and that essen tially a v oid the C n 's not only for p ositiv e times, but also for negativ e ones. The p enetration in the sets C n for n 6 = 0 annot b e made quite as small as in Prop osition 4.10. Corollary 4.11 L et X b e a pr op er ge o desi CA T( − 1) metri sp a e. L et ǫ 0 in R ∗ + ∪ {∞} , δ 0 , κ 0 ≥ 0 . L et C 0 b e a pr op er ǫ 0 - onvex subset of X , and let f 0 : [ ξ ∈ ∂ ∞ X − ∂ ∞ C 0 T 1 ξ X → [0 , + ∞ ] b e a  ontinuous map suh that f 0 | T 1 ξ 0 X is a κ 0 -p enetr ation map in C 0 for every ξ 0 ∈ ∂ ∞ X − ∂ ∞ C 0 . L et h ′ 0 ≥ h 0 = h 0 ( ǫ 0 , δ 0 , κ 0 ) , h ≥ h ′ 1 = h ′ 1 ( ǫ 0 , δ 0 , h ′ 0 ) , and h ′′ 1 = h ′ 1 ( ǫ 0 , δ 0 , h ′ 0 ) + c ′ 3 ( ǫ 0 )( δ 0 + c 1 ) + c ′ 1 ( ǫ 0 ) Assume that ther e exists a ge o desi line γ 0 in X with f 0 ( γ 0 ) = h . F or every n in N − { 0 } , let C n b e an ǫ 0 - onvex subset of X , suh that ( C n ) n ∈ N satises the assertions ( iii ) and ( iv ) with r esp e t to every ξ 0 ∈ ∂ ∞ X − ∂ ∞ C 0 . Then ther e exists a ge o desi line γ ′ in X entering C 0 at time t = 0 with f 0 ( γ ′ ) = h , suh that ℓ C n ( γ ′ ) ≤ h ′′ 1 for every n in N − { 0 } . Pro of. Let γ 0 b e a geo desi line in X with f 0 ( γ 0 ) = h , and let ξ b e the starting p oin t at innit y of γ 0 , whi h do es not b elong to ∂ ∞ C 0 as h < ∞ . Applying Prop osition 4.10 with ξ 0 = ξ , as h ≥ h ′ 1 , there exists a geo desi line γ starting from ξ and en tering C 0 at time 0 , su h that f 0 ( γ ) = h and ℓ C n ( γ ) ≤ h ′ 1 for ev ery n ∈ N − { 0 } su h that γ (] δ 0 , + ∞ [) meets C n . Let ξ ′ b e the other endp oin t at innit y of γ , whi h do es not b elong to ∂ ∞ C 0 as h < ∞ . Applying Prop osition 4.10 again with no w ξ 0 = ξ ′ , w e get that there exists a geo desi line γ ′ starting from ξ ′ and en tering C 0 at time 0 , su h that f 0 ( γ ′ ) = h and ℓ C n ( γ ′ ) ≤ h ′ 1 for ev ery n ∈ N − { 0 } su h that γ ′ (] δ 0 , + ∞ [) meets C n . Assume b y absurd that there exists n ∈ N − { 0 } su h that ℓ C n ( γ ′ ) > h ′′ 1 > 0 . Then γ en ters C n at a p oin t x ′ n , exiting it at a p oin t y ′ n at time at most δ 0 , as h ′′ 1 > h ′ 1 b y the denition of h ′′ 1 . In partiular, if x ′ = γ ′ (0) is the en tering p oin t of γ ′ in C 0 , then d ( y ′ n , x ′ ) ≤ δ 0 if x ′ , y ′ n , x ′ n , ξ ′ are not in this order on γ ′ . 54 x ′ x ′ n y ′ n y γ ξ γ ′ ξ ′ C 0 y n x n C n Let y b e the exiting p oin t of γ out of C 0 . Note that h ≥ h ′ 1 ≥ h ′ 0 ≥ h 0 ≥ h ′ ( ǫ 0 , sinh( δ 0 + c 1 )) ≥ h ′ ( ǫ 0 , δ 0 + c 1 ) (- 30 -) b y the denitions of h ′ 1 , h 0 , h ′ . By Lemma 2.5 if ǫ 0 6 = ∞ and b y Lemma 2.11 if ǫ 0 = ∞ and ( f 0 , δ 0 ) 6 = ( ph C 0 , 0) , and as in the pro of of Lemma 4.6 if ( ǫ 0 , f 0 , δ 0 ) = ( ∞ , ph C 0 , 0) sine h ≥ h 0 , w e ha v e d ( x ′ , y ) ≤ c 1 . Hene b y on v exit y , d ( y ′ n , γ ) ≤ d ( x ′ , γ ) + δ 0 ≤ d ( x ′ , y ) + δ 0 ≤ δ 0 + c 1 . Note that d ( x ′ n , y ′ n ) = ℓ C n ( γ ′ ) > h ′′ 1 ≥ h ′ 1 ≥ h ′ ( ǫ 0 , δ 0 + c 1 ) b y the denition of h ′′ 1 and b y the inequalities (- 30 - ). Hene, b y Lemma 2.8 if ǫ 0 6 = ∞ and b y Lemma 2.14 otherwise, the geo desi line γ en ters C n at a p oin t x n and exits it at a p oin t y n su h that d ( y ′ n , x n ) ≤ c ′ 3 ( ǫ 0 ) d ( y ′ n , γ ) or d ( x n , y n ) ≥ d ( x ′ n , y ′ n ) . (- 31 -) Let us pro v e b y absurd that γ (] δ 0 , + ∞ [) meets C n . Otherwise, sine γ − 1 ( y ) ≥ 0 , b y on v exit y , and b y Lemma 2.5 if ǫ 0 6 = ∞ or Lemma 2.11 if ǫ 0 = ∞ , w e ha v e d ( x ′ , x ′ n ) ≤ d ( x ′ , y ) + δ 0 + d ( y n , x ′ n ) ≤ c 1 + δ 0 + c ′ 1 ( ǫ 0 ) . (- 32 -) By the inequalities (- 30 - ) and b y the denition of h ′ ( ǫ, η ) , w e ha v e h ′ 1 ≥ h ′ ( ǫ 0 , sinh( δ 0 + c 1 )) ≥ 2 sinh( δ 0 + c 1 ) ≥ 2 δ 0 + c 1 . Hene the inequalities (- 32 - ) on tradits the fat that, b y the denition of h ′′ 1 , d ( x ′ , x ′ n ) ≥ d ( x ′ n , y ′ n ) − δ 0 > h ′′ 1 − δ 0 ≥ h ′ 1 − δ 0 + c ′ 1 ( ǫ 0 ) ≥ c 1 + δ 0 + c ′ 1 ( ǫ 0 ) . Assume that the seond of the inequalities (- 31 - ) holds true. As d ( x ′ n , y ′ n ) > h ′ 1 , this on tradits the onstrution of γ . Hene w e ha v e d ( y ′ n , x n ) ≤ c ′ 3 ( ǫ 0 ) d ( y ′ n , γ ) ≤ c ′ 3 ( ǫ 0 )( δ 0 + c 1 ) . But then, b y the triangular inequalit y and b y the denition of h ′′ 1 , d ( x n , y n ) ≥ d ( x ′ n , y ′ n ) − d ( x n , y ′ n ) − d ( y n , x ′ n ) > h ′′ 1 − c ′ 3 ( ǫ 0 )( δ 0 + c 1 ) − c ′ 1 ( ǫ 0 ) = h ′ 1 , whi h on tradits the onstrution of γ .  55 5 Presribing the p enetration of geo desi lines In this setion, w e apply Prop osition 4.10 to pro v e a n um b er of results on the geo desi o w of negativ ely urv ed Riemannian manifolds. The follo wing onstan ts app ear in the theorems, dep ending on ǫ ∈ R ∗ + ∪ {∞} , δ , κ ≥ 0 . • c ′′ 1 = c ′′ 1 ( ǫ, δ , κ ) = max  2 c ′ 1 ( ǫ ) + 2 δ + κ, h ′ 1 ( ǫ, δ , h 0 ( ǫ, δ , c ′ 1 ( ∞ ))  . • c ′′ 2 ( ǫ ) = c ′′ 1 ( ǫ, 0 , 0) + c ′ 1 ( ∞ ) + 2 c 1 , where c 1 = c ′ 1 ( ǫ ) if ǫ 6 = ∞ , and c 1 = 1 / 19 otherwise. Note that c ′′ 2 ( ∞ ) = h ′ 1 ( ∞ , 0 , h 0 ( ∞ , 0 , c ′ 1 ( ∞ )) + c ′ 1 ( ∞ ) + 2 c 1 ≈ 8 . 3712 b y the denition of c ′′ 1 and the appro ximation (- 13 - ) . Reall that the onstan ts c ′ 1 ( ǫ ) are giv en b y Lemmas 2.5 and 2.11 , and that h 0 ( · , · , · ) and h ′ 1 ( · , · , · ) are giv en in the list of onstan ts in the b eginning of the subsetion 4.2. 5.1 Clim bing in balls and horoballs In this subsetion, w e onstrut geo desi ra ys or lines ha ving presrib ed p enetration prop- erties in a ball or a horoball, while essen tially a v oiding a family of almost disjoin t on v ex subsets. Let us onsider the p enetration heigh t and inner pro jetion p enetration maps rst in horoballs and then in balls. Note that if C 0 is a ball or an horoball, if f 0 = ph C 0 , then k f 0 − ph C 0 k ∞ = 0 and if f 0 = ipp C 0 , then k f 0 − p h C 0 k ∞ ≤ c ′ 1 ( ∞ ) b y Setion 3.1 . Theorem 5.1 L et ǫ ∈ R ∗ + ∪ {∞} , δ , κ ≥ 0 ; let X b e a  omplete simply  onne te d R ie- mannian manifold with se tional urvatur e at most − 1 and dimension at le ast 3 ; let ξ 0 ∈ X ∪ ∂ ∞ X ; let C 0 b e a hor ob al l suh that ξ 0 / ∈ C 0 ∪ ∂ ∞ C 0 ; let f 0 = ph C 0 or f 0 = ipp C 0 ; let ( C n ) n ∈ N −{ 0 } b e a family of ǫ - onvex subsets of X ; for every n ∈ N − { 0 } suh that ξ 0 / ∈ C n ∪ ∂ ∞ C n , let f n : T 1 ξ 0 X → [0 , + ∞ ] b e a κ -p enetr ation map in C n . If diam( C n ∩ C m ) ≤ δ for al l n, m in N with n 6 = m , then, for every h ≥ c ′′ 1 ( ǫ, δ , k f 0 − ph C 0 k ∞ ) , ther e exists a ge o desi r ay or line γ starting fr om ξ 0 and entering C 0 at time 0 , suh that f 0 ( γ ) = h and f n ( γ ) ≤ c ′′ 1 ( ǫ, δ , k f 0 − ph C 0 k ∞ ) + κ for every n ≥ 1 suh that γ  ] δ , + ∞ [  me ets C n . Pro of. let h ≥ c ′′ 1 . In order to apply Prop osition 4.10 , dene ǫ 0 = ǫ, δ 0 = δ, κ 0 = 2 log (1 + √ 2) = c ′ 1 ( ∞ ) and h ′ 0 = h 0 ( ǫ 0 , δ 0 , κ 0 ) . Reall that ph C 0 and ipp C 0 are κ 0 -p enetration maps for C 0 b y Lemma 3.3 . F or ev ery n ∈ N − { 0 } su h that ξ 0 / ∈ C n ∪ ∂ ∞ C n , let us apply Prop osition 3.7 Case (1) to C = C 0 , C ′ = C n , f = f 0 , f ′ = ℓ C n , h ′ = h ′ 0 , so that h min = 2 c ′ 1 ( ǫ ) + 2 δ + k f 0 − p h C 0 k ∞ and h min 0 = 2 δ . Note that h min 0 ≤ h ′ 0 , as h ′ 0 ≥ h ′ ( ǫ, sinh( δ + c 1 )) ≥ 2 sinh( δ + c 1 ) ≥ 2 δ , b y the denition of h 0 and of h ′ ( · , · ) . As h ≥ c ′′ 1 ≥ h min b y the denition of c ′′ 1 , Prop osition 3.7 (1) hene implies that ( C n ) n ∈ N satises the Lo al presription prop ert y ( iv ) . Th us b y Prop osition 4.10 , there exists a geo desi ra y or line γ starting at ξ 0 su h that f 0 ( γ ) = h and ℓ C n ( γ ) ≤ c ′′ 1 , whi h implies that f n ( γ ) ≤ c ′′ 1 + κ , for ev ery n ≥ 1 su h that γ (] δ, + ∞ [) meets C n .  The pro of of the orresp onding result when C 0 is a ball of radius R ≥ ǫ is the same, using Case (2) of Prop osition 4.10 instead of Case (1). This requires h ≤ h max = 2 R − 2 c ′ 1 ( ǫ ) − k f 0 − ph C 0 k ∞ . T o b e nonempt y , the follo wing result requires R ≥ c ′′ 1 ( ǫ, δ , k f 0 − ph C 0 k ∞ ) / 2 + c ′ 1 ( ǫ ) + k f 0 − ph C 0 k ∞ / 2 . 56 Theorem 5.2 L et ǫ > 0 , δ , κ ≥ 0 ; let X b e a  omplete simply  onne te d R iemannian manifold with se tional urvatur e at most − 1 and dimension at le ast 3 ; let C 0 b e a b al l of r adius R ≥ ǫ ; let ξ 0 ∈ X ∪ ( ∂ ∞ X ) − C 0 ; let f 0 = p h C 0 or f 0 = ipp C 0 ; let ( C n ) n ∈ N −{ 0 } b e a family of ǫ - onvex subsets of X ; for every n ∈ N − { 0 } suh that ξ 0 / ∈ C n ∪ ∂ ∞ C n , let f n : T 1 ξ 0 X → [0 , + ∞ ] b e a κ -p enetr ation map C n . If diam( C n ∩ C m ) ≤ δ for al l n, m in N with n 6 = m , then, for every h ∈ h c ′′ 1  ǫ, δ, k f 0 − ph C 0 k ∞  , 2 R − 2 c ′ 1 ( ǫ ) − k f 0 − ph C 0 k ∞ i , ther e exists a ge o desi r ay or line γ starting fr om ξ 0 and entering C 0 at time 0 , suh that f 0 ( γ ) = h and f n ( γ ) ≤ c ′′ 1  ǫ, δ, k f 0 − ph C 0 k ∞  + κ for every n ≥ 1 suh that γ (] δ, + ∞ [) me ets C n .  V arying the family ( C n ) n ∈ N −{ 0 } of ǫ -on v ex subsets app earing in Theorems 5.1 and 5.2, among balls of radius at least ǫ , horoballs, ǫ -neigh b ourho o ds of totally geo desi subspaes, et, w e get sev eral orollaries. W e will only state t w o of them, Corollaries 5.3 and 5.5, whi h ha v e appliations to equiv arian t families. The pro ofs of these results are simplied v ersions of the pro of of Corollary 4.11 , giving b etter (though v ery probably not optimal) onstan ts. Corollary 5.3 L et X b e a  omplete simply  onne te d R iemannian manifold with se tional urvatur e at most − 1 and dimension at le ast 3 , and let  H n  n ∈ N b e a family of hor ob al ls in X with disjoint interiors. Then, for every h ≥ c ′′ 1 ( ∞ , 0 , 0) ≈ 6 . 5032 , ther e exists a ge o desi line γ ′ suh that ph H 0 ( γ ′ ) = h and ph H n ( γ ′ ) ≤ c ′′ 2 ( ∞ ) ≈ 8 . 371 2 for every n ≥ 1 . Pro of. Let C 0 = H 0 and let ξ b e a p oin t in ∂ ∞ X − ∂ ∞ C 0 . W e apply Theorem 5.1 with ǫ = ∞ , δ = 0 , κ = 0 , ξ 0 = ξ , C n = H n for ev ery n in N , f 0 = ph C 0 , and f n = ℓ C n for ev ery n 6 = 0 su h that ξ 0 / ∈ C n ∪ ∂ ∞ C n . Note that for ev ery n ∈ N , f n is a κ -p enetration map in C n . As h ≥ c ′′ 1 ( ǫ, 0 , 0) , there exists a geo desi line γ starting from ξ and en tering C 0 at time 0 , su h that ph C 0 ( γ ) = h and ℓ C n ( γ ) ≤ c ′′ 1 ( ǫ, 0 , 0) for ev ery n ∈ N − { 0 } su h that γ meets C n at a p ositiv e time. Let ξ ′ b e the other endp oin t of γ . This p oin t is not in ∂ ∞ C 0 . Applying Theorem 5.1 again, as ab o v e exept that no w ξ 0 = ξ ′ , w e get that there exists a geo desi line γ ′ starting from ξ ′ and en tering C 0 at time 0 , su h that ph C 0 ( γ ′ ) = h and ℓ C n ( γ ′ ) ≤ c ′′ 1 ( ǫ, 0 , 0) for ev ery n ∈ N − { 0 } su h that γ ′ meets C n at a p ositiv e time. Let c ′′ 2 = c ′′ 2 ( ǫ ) . Assume b y absurd that there exists n ∈ N − { 0 } su h that ph C n ( γ ′ ) > c ′′ 2 > 0 . Then γ ′ en ters C n at the p oin t x ′ n , exiting it at the p oin t y ′ n at a nonp ositiv e time, as c ′′ 2 > c ′′ 1 ( ǫ, 0 , 0) . In partiular, if x ′ = γ ′ (0) is the en tering p oin t of γ ′ in C 0 , then x ′ , y ′ n , x ′ n , ξ ′ are in this order on γ ′ (see the piture in the pro of of Corollary 4.11 ). Let y b e the exiting p oin t of γ out of H 0 . With c 1 = 1 / 19 , as in the pro of of Lemma 4.6 , sine ph C 0 ( γ ) and ph C 0 ( γ ′ ) are equal to h ≥ c ′′ 1 ( ∞ , 0 , 0) ≥ h ′ 1 ( ∞ , 0 , h 0 ( ∞ , 0 , c ′ 1 ( ∞ ))) ≥ h 0 ( ∞ , 0 , c ′ 1 ( ∞ )) = h ′ ( ∞ , sinh c 1 ) b y the denition of c ′′ 1 , h ′ 1 , h 0 , w e ha v e d ( x ′ , y ) ≤ c 1 . Let ξ n b e the p oin t at innit y of H n . Let p ′ b e the p oin t in [ x ′ n , y ′ n ] the losest to ξ n , so that d ( p ′ , y ′ n ) ≥ β ξ n ( y ′ n , p ′ ) = ph C n ( γ ′ ) / 2 > c ′′ 2 / 2 . 57 Let p b e the p oin t of γ the losest to p ′ . As, b y on v exit y and the denition of c ′′ 2 , w e ha v e d ( p ′ , p ) = d ( p ′ , γ ) ≤ d ( x ′ , γ ) ≤ d ( x ′ , y ) ≤ c 1 < c ′′ 2 / 2 , it follo ws that p b elongs to the in terior of C n . If p ∈ ] ξ , y ] , then b y on v exit y , c ′′ 2 / 2 < d ( p ′ , y ′ n ) ≤ d ( p ′ , x ′ ) ≤ d ( p ′ , p ) + d ( y, x ′ ) ≤ 2 c 1 , a on tradition, as b y the denition of c ′′ 2 , of c ′′ 1 and of h ′ ( ǫ, η ) (see the equations (- 4 - ) and (- 9 - )), w e ha v e c ′′ 2 ≥ c ′′ 1 ( ǫ, 0 , 0) + 2 c 1 ≥ h ′ ( ǫ, sinh c 1 ) + 2 c 1 ≥ 2 sinh c 1 + 2 c 1 > 4 c 1 . Hene p ∈ ] y , ξ ′ [ ⊂ γ (]0 , + ∞ [) , so that γ meets C n at a p ositiv e time. But, b y Prop osition 3.3 and the denition of c ′′ 2 , ℓ C n ( γ ) ≥ ph C n ( γ ) − c ′ 1 ( ∞ ) ≥ 2 β ξ n ( y ′ n , p ) − c ′ 1 ( ∞ ) ≥ 2( β ξ n ( y ′ n , p ′ ) − d ( p, p ′ )) − c ′ 1 ( ∞ ) > 2( c ′′ 2 / 2 − c 1 ) − c ′ 1 ( ∞ ) = c ′′ 1 ( ǫ, 0 , 0) . This on tradits the onstrution of γ .  Let M b e a omplete nonelemen tary geometrially nite Riemannian manifold with setional urv ature at most − 1 (see for instane [Bo w ℄ for a general referene). Reall that a usp of M is an asymptoti lass of minimizing geo desi ra ys in M along whi h the injetivit y radius on v erges to 0 . If M has nite v olume, then the set of usps of M is in bijetion with the (nite) set of ends of M , b y the map whi h asso iates to a represen tativ e of a usp the end of M to w ards whi h it on v erges. Let π : f M → M b e a univ ersal Riemannian o v ering of M , with o v ering group Γ . If e is a usp of M , and ρ e a minimizing geo desi ra y in the lass e , as M is geometrially nite and nonelemen tary , there exists (see for instane [BK, Bo w , HP5 ℄) a (unique) maximal horoball H e in f M en tered at the p oin t at innit y ξ e of a xed lift of ρ e in f M , su h that γ H e and H e ha v e disjoin t in teriors if γ ∈ Γ do es not x ξ e . The image V e of H in M is alled the maximal Mar gulis neighb orho o d of e . If ρ e starts from a p oin t in the image b y π of the horosphere b ounding H , then let h t e : M → R b e map dened b y h t e ( x ) = lim t →∞ ( t − d ( ρ e ( t ) , x )) , alled the height funtion with r esp e t to e . Let maxh t e : T 1 M → R b e dened b y maxh t e ( γ ) = su p t ∈ R h t e ( γ ( t )) . The maximum height sp e trum of the pair ( M , e ) is the subset of ] − ∞ , + ∞ ] dened b y MaxSp( M , e ) = maxht e ( T 1 M ) . Corollary 5.4 L et M b e a  omplete, nonelementary ge ometri al ly nite R iemannian man- ifold with se tional urvatur e at most − 1 and dimension at le ast 3 , and let e b e a usp of M . Then MaxSp( M , e )  ontains [ c ′′ 2 / 2 , + ∞ ] . 58 Note that c ′′ 2 / 2 ≈ 4 . 18 56 , hene Theorem 1.2 of the in tro dution follo ws. Pro of. With the ab o v e notations, let ( H n ) n ∈ N b e the Γ -equiv arian t family of horoballs in f M with pairwise disjoin t in teriors su h that H 0 = H e . Apply Corollary 5.3 to this family to get, for ev ery h ≥ c ′′ 2 ≥ c ′′ 1 ( ∞ , 0 , 0) , a geo desi line e γ in f M with ph H 0 ( e γ ) = h and ph H n ( e γ ) ≤ c ′′ 2 for ev ery n ≥ 1 . Let γ b e the lo ally geo desi line in M image b y π of e γ . Observ e that h t e ◦ π = β H n in H n and that ph H n ( e γ ) = 2 sup t ∈ R β H n ( e γ ( t )) (see Setion 3.1). Hene sup t ∈ R h t e ( γ ( t )) = h/ 2 and the result follo ws.  S hmidt and Sheingorn [SS℄ treated the ase of t w o-dimensional manifolds of on- stan t urv ature − 1 (h yp erb oli surfaes) with a usp. They sho w ed that in that ase MaxSp( M , e ) on tains the in terv al [log 100 , + ∞ ] ≈ [4 . 61 , + ∞ ] . This pap er [SS℄ w as a starting p oin t of our in v estigations, although the metho d w e use is quite dieren t from theirs. Let P b e a (neessarily nite) nonempt y set of usps of M . F or ev ery e in P ,  ho ose a horoball H e , with p oin t at innit y ξ e as ab o v e Corollary 5.4 . The horoballs of the family ( g H e ) g ∈ Γ / Γ ξ e , e ∈ P ma y ha v e non disjoin t in teriors. But as M is geometrially nite and nonelemen tary , there exists (see [BK, Bo w ℄) t ≥ 0 su h that t w o distint elemen ts in ( g H e [ t ]) g ∈ Γ / Γ ξ e , e ∈ P ha v e disjoin t in teriors. Let t P b e the lo w er b ound of all su h t 's. F or ev ery γ ∈ T 1 M , dene maxh t P ( γ ) = m ax e ∈ P maxh t e ( γ ) and MaxSp( M , P ) = maxht P ( T 1 M ) . Remark. Let C b e the set of all usps of M . Under the same h yp otheses as in Corollary 5.4 , the follo wing t w o assertions hold, b y applying Corollary 5.3 to the family of horoballs ( g H ′ e ′ ) g ∈ Γ / Γ ξ e ′ , e ′ ∈ C with H ′ e ′ = H e if e ′ = e , and H ′ e ′ = H e ′ [ t ] for some t big enough otherwise, for the rst assertion, and to the family ( g H e [ t P ]) g ∈ Γ / Γ ξ e , e ∈ P for the seond one. (1) F or ev ery usp e of M , there exists a onstan t t ≥ 0 su h that for ev ery h ≥ t , there exists a lo ally geo desi line γ in M su h that maxh t e ( γ ) = h and maxh t e ′ ( γ ) ≤ t for ev ery usp e ′ 6 = e in M . (2) Let P b e a nonempt y set of usps of M . Then MaxSp( M , P ) on tains the haline [( c ′′ 2 + t P ) / 2 , + ∞ ] . No w, w e pro v e the analogs of Corollaries 5.3 and 5.4 for families of balls with disjoin t in teriors. Let R min 0 = 7 sinh c ′ 1 ( ∞ ) + 3 2 c ′ 1 ( ∞ ) ≈ 22 . 44 31 . Corollary 5.5 L et X b e a  omplete simply  onne te d R iemannian manifold with se tional urvatur e at most − 1 and dimension at le ast 3 , and let  B n  n ∈ N b e a family of b al ls in X with disjoint interiors suh that the r adius R 0 of B 0 is at le ast R min 0 . F or every h ∈  c ′′ 1 ( R min 0 , 0 , 0) , 2 R 0 − 2 c ′ 1 ( R min 0 )  , ther e exists a ge o desi line γ in X with ph B 0 ( γ ) = h and ph B n ( γ ) ≤ c ′′ 2 ( R min 0 ) for al l n ≥ 1 . Pro of. W e start b y some omputations. Let ǫ > 0 . With c 1 = c ′ 1 ( ǫ ) and c 5 = c 5 ( ǫ, 0) as in Subsetion 4.2 , w e ha v e c 5 ≥ 6 s in h c 1 sine c ′ 3 ( ǫ ) ≥ 3 b y the denition of c ′ 3 ( ǫ ) in Equation (- 5 - ). By the denition of h 0 in Subsetion 4.2 and of h ′ in Equation (- 4 -) , w e 59 ha v e h 0 ( ǫ, 0 , c ′ 1 ( ∞ )) ≥ h ′ ( ǫ, sinh c 1 ) ≥ 2 sinh c 1 . Hene, b y the denition of c ′′ 2 , c ′′ 1 , h ′ 1 and as ǫ 7→ c ′ 1 ( ǫ ) is dereasing, c ′′ 2 ( ǫ ) = c ′′ 1 ( ǫ, 0 , 0) + c ′ 1 ( ∞ ) + 2 c 1 = max { 2 c 1 , h 0 ( ǫ, 0 , c ′ 1 ( ∞ )) + 2 c 5 } + c ′ 1 ( ∞ ) + 2 c 1 ≥ 2 sinh c 1 + 1 2 s inh c 1 + c ′ 1 ( ∞ ) + 2 c 1 ≥ 14 sinh c ′ 1 ( ∞ ) + 3 c ′ 1 ( ∞ ) . Dene no w ǫ = R min 0 , so that 2 ǫ ≤ c ′′ 2 ( ǫ ) and R 0 ≥ ǫ . F or ev ery n 6 = 0 , let R n b e the radius of the ball B n . If for some n 6 = 0 w e ha v e 2 R n ≤ c ′′ 2 ( ǫ ) , then ph B n ( γ ) ≤ c ′′ 2 ( ǫ ) and the last assertion of Corollary 5.5 holds for this n . Hene up to remo ving balls, w e ma y assume that R n ≥ c ′′ 2 ( ǫ ) / 2 ≥ ǫ for ev ery n 6 = 0 , so that the balls in ( B n ) n ∈ N are ǫ -on v ex. The end of the pro of is no w exatly as the pro of of Corollary 5.3 , with the follo wing mo diations: ξ is an y p oin t in ∂ ∞ X ; ǫ = R min 0 ; C n = B n for ev ery n in N ; w e apply Theorem 5.2 instead of Theorem 5.1 , whi h is p ossible b y the range assumption on h ; w e tak e no w c 1 = c ′ 1 ( ǫ ) , so that w e still ha v e d ( x ′ , y ) ≤ c 1 b y Prop osition 2.5; ξ n is no w the en ter of B n , and β ξ n ( u, v ) = d ( u, ξ n ) − d ( v, ξ n ) (see Setion 2.1 ). Besides that, the pro of is un hanged.  A hea vy omputation sho ws that c ′ 1 ( R min 0 ) ≈ 1 . 7627 , c ′′ 1 ( R min 0 , 0 , 0) ≈ 101 . 4169 and c ′′ 2 ( R min 0 ) ≈ 106 . 70 51 . Note that the ab o v e orollary is nonempt y only if R 0 ≥ c ′′ 1 ( R min 0 , 0 , 0) / 2 + c ′ 1 ( R min 0 ) ≈ 52 . 471 2 . The onstan ts in the follo wing orollary are not optimal. Theorem 1.3 in the in tro dution follo ws from it. Corollary 5.6 L et M b e a  omplete R iemannian manifold with se tional urvatur e at most − 1 and dimension at le ast 3 , let ( x i ) i ∈ I b e a nite or  ountable family of p oints in M with r i = inj M x i , suh that d ( x i , x j ) ≥ r i + r j if i 6 = j and suh that r i 0 ≥ 56 for some i 0 ∈ I . Then, for every d ∈ [2 , r i 0 − 54] , ther e exists a lo  al ly ge o desi line γ p assing at distan e exatly d fr om x i 0 at time 0 , r emaining at distan e gr e ater than d fr om x i 0 at any nonzer o time, and at distan e at le ast r i − 56 fr om x i for every i 6 = i 0 . In p artiular, min t ∈ R d ( γ ( t ) , x i 0 ) = d. Pro of. Let π : f M → M b e a univ ersal o v ering of M , with o v ering group Γ , and x a lift e x i of x i for ev ery i ∈ I . Let B i b e the ball B f M ( e x i , r i ) . Apply Corollary 5.5 to the family of balls ( g B i ) g ∈ Γ , i ∈ I in X = f M , whi h ha v e pairwise disjoin t in teriors b y the denition of r i and the assumption on d ( x i , x j ) . Note that r i 0 ≥ 56 ≥ R min 0 (see the denition of R min 0 ). Let h = 2( r i 0 − d ) , whi h b elongs to [108 , 2 r i 0 − 4] , whi h is on tained in [ c ′′ 1 ( R min 0 , 0 , 0) , 2 r i 0 − 2 c ′ 1 ( R min 0 )] b y the previous omputations. Then Corollary 5.5 implies that there exists a geo desi line e γ in f M su h that ph B i 0 ( γ ) = h and ph g B i ( γ ) ≤ c ′′ 2 ( R min 0 ) < 108 for all ( g , i ) 6 = (1 , i 0 ) . P arametrize e γ su h that its losest p oin t to e x i 0 is at time t = 0 . Let γ = π ◦ e γ , then the result follo ws b y the denition of ph C (see Subsetion 3.1).  5.2 Spiralling aroung totally geo desi subspaes In this subsetion, w e apply Prop osition 4.10 and Corollary 4.11 when C 0 is a tubular neigh b orho o d of a totally geo desi submanifold. W e only giv e a few of the v arious p ossible appliations, others an b e obtained b y v arying the ob jets ( C n ) n ∈ N −{ 0 } , as w ell as the v arious sub ases in Prop osition 3.7 (3) and (4). 60 Theorem 5.7 L et ǫ > 0 , δ ≥ 0 . L et X b e a  omplete simply  onne te d R iemannian manifold with se tional urvatur e at most − 1 and dimension at le ast 3 . L et L b e a  omplete total ly ge o desi submanifold of X with dimension at le ast 2 , dier ent fr om X , and C 0 = N ǫ L . L et ( C n ) n ∈ N −{ 0 } b e a family of ǫ - onvex subsets in X suh that diam( C n ∩ C m ) ≤ δ for al l n 6 = m in N . L et either f 0 = ftp L or f 0 = ℓ N ǫ L , with X having  onstant urvatur e in this se  ond  ase. L et h ′ 0 = h 0  ǫ 0 , δ 0 , max {k f 0 − ℓ N ǫ L k ∞ , k f 0 − ftp L k ∞ + 2 ǫ − 8 c ′ 1 ( ǫ ) }  and h ≥ h ′ 1 = h ′ 1 ( ǫ, δ , h ′ 0 ) . • F or every ξ ∈ ( X ∪ ∂ ∞ X ) − ( C 0 ∪ ∂ ∞ C 0 ) , ther e exists a ge o desi r ay or line γ starting fr om ξ and entering N ǫ L at time 0 with f 0 ( γ ) = h , and with ℓ C n ( γ ) ≤ h ′ 1 for every n 6 = 0 suh that γ (] δ, + ∞ [) me ets C n . • Ther e exists a ge o desi line γ in X with f 0 ( γ ) = h , and with ℓ C n ( γ ) ≤ h ′ 1 + c ′ 3 ( ǫ )  δ + c ′ 1 ( ǫ )  + c ′ 1 ( ǫ ) for al l n 6 = 0 . Note that if ℓ C n ( γ ) ≤ c , then f ( γ ) ≤ c + κ for an y κ -p enetration map f in C n . Pro of. W e apply Prop osition 4.10 and Corollary 4.11 with ǫ 0 = ǫ , δ 0 = δ , κ 0 = max {k f 0 − ℓ N ǫ L k ∞ , k f 0 − ftp L k ∞ + 2 ǫ − 8 c ′ 1 ( ǫ ) } , so that h ′ 0 = h 0 ( ǫ 0 , δ 0 , κ 0 ) , and f 0 is a on tin uous κ 0 -p enetration map in C 0 . As L is a omplete totally geo desi submanifold of dimension and o dimension at least 1 , there do es exist a geo desi line γ 0 in X su h that f 0 ( γ 0 ) = h . Let h min 0 = δ 0 and h min = 4 c ′ 1 ( ǫ ) + 2 ǫ + δ + k f 0 − ftp L k ∞ . By the denitions of h ′ 1 ( · , · , · ) , h 0 ( · , · , · ) , c 5 ( · , · ) in Subsetion 4.2 , w e ha v e h ′ 0 = h 0 ( ǫ 0 , δ 0 , κ 0 ) > δ 0 = h min 0 , and h ≥ h ′ 1 = h 0 ( ǫ 0 , δ 0 , κ 0 ) + 2 c 5 ( ǫ 0 , δ 0 ) ≥ κ 0 + 12 sin h( c ′ 1 ( ǫ 0 ) + δ 0 ) ≥ κ 0 + 12 c ′ 1 ( ǫ ) + δ ≥ h min . The family ( C n ) n ∈ N hene satises the Lo al presription prop ert y ( iv ) b y Prop osition 3.7 (3). Therefore, the result follo ws from Prop osition 4.10 and Corollary 4.11  Remark 5.8 If the C n 's are disjoin t from N ǫ L (and δ = 0 ), then the same result as Theorem 5.7 also holds when L has dimension 1 , b y replaing Prop osition 3.7 (3) b y Prop osition 3.7 (4) in the ab o v e pro of and h min 0 = δ 0 b y h min 0 = 0 . Theorem 5.9 L et ǫ > 0 , δ ≥ 0 . L et X b e a  omplete simply  onne te d R iemannian manifold with se tional urvatur e at most − 1 and dimension at le ast 3 . L et ( L n ) n ∈ N b e a family of ge o desi lines in X , suh that diam( N ǫ L n ∩ N ǫ L m ) ≤ δ for al l n 6 = m in N . L et either f 0 = ftp L 0 , or f 0 = ℓ N ǫ L 0 if X has  onstant urvatur e, or f 0 = crp L 0 if the metri spher es for the Hamenstädt distan es (on ∂ ∞ X − { ξ } for any ξ ∈ ∂ ∞ X ) ar e top olo gi al spher es. L et h ′ 0 = max { 5 c ′ 1 ( ǫ ) + 5 ǫ + δ , h 0 ( ǫ, δ , max {k f 0 − ℓ N ǫ L 0 k ∞ , k f 0 − ftp L 0 k ∞ + 2 ǫ − 8 c ′ 1 ( ǫ ) } ) } and h ≥ h ′ 1 = h ′ 1 ( ǫ, δ , h ′ 0 ) . 61 • F or every ξ ∈ ( X ∪ ∂ ∞ X ) − ( N ǫ L 0 ∪ ∂ ∞ L 0 ) (and ξ ∈ ∂ ∞ X − ∂ ∞ L 0 if f 0 = crp L 0 ), ther e exists a ge o desi r ay or line γ starting fr om ξ and entering N ǫ L 0 at time 0 with f 0 ( γ ) = h , suh that ℓ N ǫ L n ( γ ) ≤ h ′ 1 for every n 6 = 0 suh that γ (] δ, + ∞ [) me ets N ǫ L n . • Ther e exists a ge o desi line γ in X suh that f 0 ( γ ) = h , and, if n 6 = 0 , then ℓ N ǫ L n ( γ ) ≤ h ′ 1 + c ′ 3 ( ǫ )  δ + c ′ 1 ( ǫ )  + c ′ 1 ( ǫ ) . Note that if ℓ N ǫ L n ( γ ) ≤ c , then f ( γ ) ≤ c + κ for an y κ -p enetration map f in N ǫ L n . Pro of. As in the previous pro of, w e apply Prop osition 4.10 and Corollary 4.11 with C n = N ǫ L n , ǫ 0 = ǫ , δ 0 = δ , κ 0 = max {k f 0 − ℓ N ǫ L 0 k ∞ , k f 0 − ftp L 0 k ∞ + 2 ǫ − 8 c ′ 1 ( ǫ ) } . F or ev ery n 6 = 0 , let h min 0 = 3 c ′ 1 ( ǫ ) + 3 ǫ + δ + k ℓ N ǫ L n − ftp L n k ∞ and h min = 4 c ′ 1 ( ǫ ) + 2 ǫ + δ + k f 0 − ftp L k ∞ . In partiular, h ′ 0 = max { 5 c ′ 1 ( ǫ ) + 5 ǫ + δ , h 0 ( ǫ, δ , κ 0 ) } ≥ h min 0 b y Lemma 3.4 . As in the end of the previous pro of, the family ( C n ) n ∈ N hene satises the prop ert y ( iv ) b y Prop osition 3.7 (4), and the result follo ws.  Let M b e a omplete Riemannian manifold with setional urv ature at most − 1 and dimension n ≥ 3 . Fix a univ ersal o v er f M → M of M . F or ǫ > 0 , δ ≥ 0 , a (p ossibly not onneted, but an y t w o omp onen ts ha ving equal dimension) immersed omplete totally geo desi submanifold L (of dimension at least 1 and at most n − 1 ) will b e alled ( ǫ, δ ) - sep ar ate d if the diameter of the in tersetion of the ǫ -neigh b ourho o ds of t w o lifts to f M of t w o omp onen ts of L is at most δ . Examples. (1) If L is ompat and em b edded, then there exists ǫ > 0 su h that L is ( ǫ, 0) -separated. F or instane, a nite family of disjoin t simple losed geo desis is ( ǫ, 0) -separated for ǫ small enough. (2) If L is ompat, and if L is self-tr ansverse (i.e. if the tangen t spaes at ev ery double p oin t of L are transv erse), then for ev ery ǫ > 0 small enough, L is ( ǫ, 1) -separated. In partiular, a nite family of losed geo desis (p ossibly nonsimple) is ( ǫ, 1) -separated for ǫ small enough. (3) The lift of a lo ally geo desi line γ : R → M to the unit tangen t bundle T 1 M is the map e γ : R → T 1 M (or b y abuse its image) giv en b y e γ ( t ) = ( γ ( t ) , γ ′ ( t )) for ev ery t ∈ R . F or ev ery ρ > 0 , if the ρ -neigh b ourho o d (for the standard Riemannian metri of T 1 M ) of the lift of γ to T 1 M is a tubular neigh b ourho o d, then there exists δ ( ρ ) ≥ 0 su h that γ is ( ρ, δ ( ρ )) -separated. Indeed, if the in tersetion of the ρ -neigh b ourho o ds of t w o dieren t lifts to a univ ersal o v er of γ has diameter big enough (dep ending only on ρ ), then b y argumen ts similar to the ones in the pro of of Lemma 2.2, t w o subsegmen ts of the t w o lifts will follo w themselv es losely for some time, hene the tangen t v etors at t w o p oin ts on these t w o lifts will b e loser than ρ . Let L b e an ( ǫ, δ ) -separated immersed omplete totally geo desi submanifold. Let ( e L α ) α ∈ A b e the family of (onneted) omplete totally geo desi submanifolds of f M , that are the lifts to f M of the omp onen ts of L . Note that in partiular, the family ( N ǫ ( e L α )) α ∈ A is lo ally nite. Let f b e one of the sym b ols ℓ, bp , f tp , crp and assume that L has dimension 1 if f = crp . Let κ f b e resp etiv ely 0 , 2 c ′ 1 ( ǫ ) , 2 c ′ 1 ( ǫ ) + 2 ǫ , 2 c ′ 1 ( ǫ ) + 2 c ′ 1 ( ∞ ) + 2 ǫ . F or ev ery lo ally geo desi 62 line γ in M , onsider a lift e γ of γ to f M . F or ev ery α ∈ A , let f α = ℓ N ǫ L α , bp L α , ftp L α , crp L α resp etiv ely , whi h is a κ f -p enetration map in N ǫ L α b y Subsetion 3.1 . The family ( f α ( e γ )) α ∈ A will b e alled the family of spir aling times of γ along L with resp et to f (and length spir aling times , fel low-tr aveling times or r ossr atio spir aling times if f = ℓ, ftp , crp resp etiv ely). Up to p erm utation of A , it do es not dep end on the  hoie of the lift e γ of γ . The en tering times of e γ in the sets N ǫ L α with f α ( e γ ) > δ + κ f , where α v aries in A , form a disrete subset (with m ultipliit y one) of R , as N ǫ L α ∩ N ǫ L β has diameter at most δ if α 6 = β . W e will only b e in terested in the orresp onding spiraling times. It is also then p ossible to order these spiraling times using the order giv en b y the parametrisation on e γ , but w e will not need this here. When L is em b edded, and ǫ is small enough so that the ǫ -neigh b orho o d of L is a tubular neigh b orho o d, then the (big enough) fello w-tra v eling times are the ones dened in the in tro dution, see the piture b elo w. γ y n x n γ ( s n ) γ ( t n ) N ǫ L L p n Corollary 5.10 L et M b e a  omplete R iemannian manifold with se tional urvatur e at most − 1 and dimension n ≥ 3 . L et ǫ > 0 , δ ≥ 0 . L et L b e an ( ǫ, δ ) -sep ar ate d immerse d  omplete total ly ge o desi submanifold (of dimension at le ast 1 and at most n − 1 ). L et f b e one of the symb ols ℓ, ftp , crp , and κ ′ f = max { 0 , 4 ǫ − 6 c ′ 1 ( ǫ ) } , 2 c ′ 1 ( ǫ ) + 2 ǫ , 2 c ′ 1 ( ǫ ) + 2 ǫ + 2 c ′ 1 ( ∞ ) r esp e tively. If f = ℓ , assume that M has  onstant urvatur e. If f = crp , assume that L has dimension 1 and that the metri spher es for the Hamenstädt distan es (on the puntur e d b oundary of a universal  over of M ) ar e top olo gi al spher es. F or every h ≥ h ′ 1 = h ′ 1  ǫ, δ, max { 5 c ′ 1 ( ǫ ) + 5 ǫ + δ , h 0 ( ǫ, δ , κ ′ f ) }  , ther e exists a lo  al ly ge o desi line γ in M having one spir aling time with r esp e t to f exatly h , and al l others b eing at most h ′ 1 + c ′ 3 ( ǫ )  δ + c ′ 1 ( ǫ )  + c ′ 1 ( ǫ ) . If furthermor e M is nonelementary and ge ometri al ly nite, then for every usp e of M , we may also assume that the lo  al ly ge o desi line γ do es not enter to o muh into the maximal Mar gulis neighb orho o d of e , i.e. γ satises maxh t e ( γ ) ≤ su p x ∈ L h t e ( x ) + ǫ + 1 2  h ′ 1 + c ′ 3 ( ǫ )( δ + c ′ 1 ( ǫ )) + c ′ 1 ( ǫ )  . Pro of. Let π : f M → M b e a univ ersal o v er of M , with o v ering group Γ . With κ 0 the onstan t in the pro ofs of the theorems 5.7 and 5.9 , it is easy to  he k, using Setion 3.1, that κ ′ f ≥ κ 0 for ev ery ase of f . The rst assertion follo ws from Theorem 5.9 applied to the family ( L n ) n of the lifts of the omp onen ts of L to f M , if the dimension of L is 1 , and from Theorem 5.7 otherwise. T o pro v e the last assertion, with the notations of Setion 5.1, let t e = su p x ∈ L h t e ( x ) + ǫ . W e add to the family of on v ex subsets in Theorem 5.7 if dim L ≥ 2 , and in the pro of of 63 Theorem 5.9 otherwise, the family of horoballs γ H e [ t e ] for γ in Γ (mo dulo the stabilizer Γ ξ e ). Note that these horoballs ha v e pairwise disjoin t in teriors, and that their in teriors are disjoin t from the ǫ -neigh b orho o d of ev ery lift of a omp onen t of L .  Theorem 1.4 in the in tro dution follo ws from this one, b y the ab o v e example (1). Remark. (1) If w e w an ted to ha v e the same lo ally geo desi line γ for ev ery usp e of M in the seond assertion of Corollary 5.10 , w e should add the bigger family of horoballs ( γ H e [ t e ]) γ ∈ Γ / Γ ξ e , e ∈ C , and replae there t e b y max e ∈ C { t e , t C } , where C is the set of usps of M , and t C is the lo w er b ound of t ≥ 0 su h that t w o distint elemen ts in ( γ H e [ t ]) γ ∈ Γ / Γ ξ e , e ∈ C ha v e disjoin t in teriors (see the denition ab o v e Corollary 5.5), in order for the new horoballs to ha v e disjoin t in teriors. (2) With M and L as ab o v e, let f b e one of the sym b ols ℓ, bp , ftp , crp . Dene, for ev ery lo ally geo desi line γ in M , maxspt L,f ( γ ) = su p α ∈ A f α ( e γ ) , the least upp er b ound of spiraling times of γ around L with resp et to f . Let MaxSp L,f ( M ) = { maxspt L,f ( γ ) : γ ∈ T 1 M } b e the maximum spir aling sp e trum MaxSp L,f ( M ) around L with resp et to f . Theo- rem 5.10 giv es, in partiular, suien t onditions for the maxim um spiraling sp etrum to on tain a ra y [ c, + ∞ ] . 5.3 Reurren t geo desis and related results In this setion, when M is geometrially nite, w e onstrut lo ally geo desi lines that ha v e a presrib ed heigh t in a usp neigh b ourho o d of M , and furthermore satisfy some reurrene prop erties. W e will use the notation in tro dued in Setion 5.1 onerning the usps e , and the ob jets h t e , V e , H e , ξ e . Corollary 5.11 L et M b e a  omplete, nonelementary, ge ometri al ly nite R iemannian manifold with  omp at total ly ge o desi b oundary, with se tional urvatur e at most − 1 and dimension at le ast 3 . L et e b e a usp of M . Then ther e exists a  onstant c ′′ 3 = c ′′ 3 ( e, M ) suh that for every h ′ ≥ c ′′ 3 , ther e exists a lo  al ly ge o desi line γ in M with maxh t e ( γ ) = h ′ , suh that the spir aling times of γ along the b oundary ∂ M ar e at most c ′′ 3 . Up to  hanging the onstan t c ′′ 3 , w e ma y also assume that γ sta ys a w a y from some xed (small enough) usp neigh b ourho o d of ev ery usp dieren t from e . Note that, up to  hanging the onstan t c ′′ 3 , the last assertion of the orollary do es not dep end on the  hoie of f = ℓ, bp , ftp , crp , with resp et to whi h the spiraling times are omputed, and w e will use f = ℓ . Pro of. As ∂ M is ompat, there exists ǫ ′ ∈ ]0 , 1[ su h that the ǫ ′ -neigh b ourho o d of the geo desi b oundary ∂ M is a tubular neigh b ourho o d of ∂ M . By denition of manifolds with totally geo desi b oundary , there exists a omplete simply onneted Riemannian manifold f M , a nonelemen tary , torsion-free, geometrially nite disrete subgroup Γ of isometries of f M , a Γ -equiv arian t olletion ( L + k ) k ∈ N of pairwise disjoin t op en halfspaes with totally 64 geo desi b oundary ( L k ) k ∈ N , su h that M is isometri with Γ \ ( f M − S k ∈ N L + k ) . W e will iden tify M and Γ \ ( f M − S k ∈ N L + k ) b y su h an isometry from no w on. Note that ( N ǫ ′ L + k ) k ∈ N is a family of pairwise disjoin t ǫ ′ -on v ex subsets in f M . Let t e,∂ M = max x ∈ ∂ M h t e ( x ) , whi h exists sine ∂ M is ompat. Note that the family ( g H e [ t e,∂ M + 1]) g ∈ Γ / Γ ξ e is a Γ -equiv arian t family of pairwise disjoin t horoballs in f M , whi h are disjoin t from N ǫ ′ L + n for all n ∈ N . Let us relab el this family of horoballs as ( H k ) k ∈ N su h that H 0 = H e [ t e,∂ M + 1] . Note that the horoballs H k , k ∈ N , are ǫ ′ -on v ex. Dene c ′′ 3 = max  h ′ 1  ǫ ′ , 0 , h 0 ( ǫ ′ , 0 , c ′ 1 ( ∞ ))  , c ′ 1 ( ǫ ′ )  + t e,∂ M + 1 + c ′ 1 ( ǫ ′ )( c ′ 3 ( ǫ ′ ) + 1 ) . and let h ′ ≥ c ′′ 3 . W e apply Corollary 4.11 with X = f M ; ǫ 0 = ǫ ′ ; δ 0 = 0 ; κ 0 = c ′ 1 ( ∞ ) ; C 0 = H 0 ; f 0 = ph C 0 ; h ′ 0 = h 0 ( ǫ 0 , δ 0 , κ 0 ) ; C 2 k + 1 = N ǫ ′ L + k ; C 2 k = H k ; h = 2 h ′ − 2( t e,∂ M + 1) . Note that f 0 is a κ 0 -p enetration map in C 0 b y Lemma 3.3 and h ≥ h ′ 1 ( ǫ 0 , δ 0 , h ′ 0 ) , as h ′ ≥ c ′′ 3 . As f M is a manifold of dimension at least 2 , there do es exist a geo desi line γ 0 in X with f 0 ( γ 0 ) = h . The family ( C n ) n ∈ N , whose elemen ts ha v e pairwise disjoin t in teriors, satises the assertion ( iii ) . It also satises ( iv ) , b y the same pro of as the one of Theorem 5.1 as h ≥ 2 c ′ 1 ( ǫ ′ ) . Hene, b y Corollary 4.11 , there exists a geo desi line e γ in X with ph H 0 ( e γ ) = h and ℓ C n ( e γ ) ≤ h ′′ 1 = h ′ 1 ( ǫ 0 , δ 0 , h ′ 0 ) + c ′ 1 ( ǫ 0 )( c ′ 3 ( ǫ 0 ) + 1 ) for all n 6 = 0 . As ℓ C 2 n +1 ( e γ ) is nite, the geo desi e γ do esn't ross the b oundary of L + n , hene sta ys in f M − S k ∈ N L + k . Let π : f M − S k ∈ N L + k → M b e the anonial pro jetion, and γ = π ◦ e γ . Hene, the length tra v eling times of γ are at most c ′′ 3 . Note that ph H e ( e γ ) = ph C 0 ( e γ ) + 2( t e,∂ M + 1) = h + 2( t e,∂ M + 1) = 2 h ′ , b y the paragraph ab o v e Lemma 3.3 . F urthermore, if g ∈ ( Γ − Γ ξ e ) / Γ ξ e , then there exists k in N − { 0 } su h that ph g H e ( e γ ) = ph C 2 k ( e γ ) + 2( t e,∂ M + 1) ≤ ℓ C 2 k ( e γ ) + c ′ 1 ( ∞ ) + 2( t e,∂ M + 1) ≤ h ′′ 1 + c ′ 1 ( ∞ ) + 2( t e,∂ M + 1) ≤ 2 c ′′ 3 ≤ 2 h ′ . Therefore maxh t e ( γ ) = h ′ b y the same pro of as in the end of the pro of of Corollary 5.4.  Let M b e a ompat, onneted, orien table, irreduible, aylindrial, atoroidal, b ound- ary inompressible 3 -manifold with nonempt y b oundary (see for instane [MT℄ for ref- erenes on 3 -manifolds and Kleinian groups). A hyp erb oli strutur e on a manifold is a omplete Riemannian metri with onstan t setional urv ature − 1 . A usp e of a h yp er- b oli struture is maximal if the maximal Margulis neigh b orho o d of e is a neigh b orho o d of an end of the manifold. Let P b e the union of the torus omp onen ts of ∂ M , and G F ( M ) = G F ( M , P ) b e the (nonempt y) spae of omplete geometrially nite h yp erb oli strutures in the in terior of M whose usps are maximal, up to isometries isotopi to the iden tit y . Reall that G F ( M ) is homeomorphi to the T ei hm üller spae of ∂ 0 M = ∂ M − P . F or ev ery σ in G F ( M ) , the usps of σ are in one-to-one orresp ondene with the torus omp onen ts of ∂ M , as an y minimizing geo desi ra y represen ting a usp on v erges to an end 65 of the in terior of M orresp onding to a torus omp onen t of ∂ M . If e is a torus omp onen t of ∂ M , let maxh t σ , e ( γ ) denotes the maxim um heigh t of a lo ally geo desi line γ in σ with resp et to the usp orresp onding to e . The  onvex  or e of a struture σ in G F ( M ) is the smallest losed on v ex subset of the in terior of M , whose injetion in the in terior of M indues an isomorphism on the fundamen tal groups. The follo wing result generalizes Theorem 1.5 in the in tro dution to the ase of sev eral usps. Corollary 5.12 L et M b e a  omp at,  onne te d, orientable, irr e duible, aylindri al, ator- oidal, b oundary in ompr essible 3 -manifold with b oundary, and let e b e a torus  omp onent of ∂ M . F or every  omp at subset K in G F ( M ) , ther e exists a  onstant c ′′ 4 = c ′′ 4 ( K ) suh that for every h ≥ c ′′ 4 and every σ ∈ K , ther e exists a lo  al ly ge o desi line γ  ontaine d in the  onvex  or e of σ suh that maxh t σ , e ( γ ) = h , and maxh t σ , e ′ ( γ ) ≤ c ′′ 4 for every torus  omp onent e ′ 6 = e of ∂ M . Pro of. F or a subset A of ∂ ∞ H 3 R , w e denote b y Con v A the h yp erb oli on v ex h ull of A in H 3 R . A subgroup Γ of π 1 M is alled a b oundary sub gr oup if there are an elemen t γ ∈ π 1 M , a omp onen t C of ∂ 0 M , and a p oin t x ∈ C su h that Γ = γ Im  π 1 ( C, x ) → π 1 ( M , x )  γ − 1 . Let (Γ n ) n ∈ N b e the olletion of b oundary subgroups of π 1 M . Let (Γ ′ n ) n ∈ N b e the olletion of maximal (rank 2 ) ab elian subgroups of π 1 M , with Γ ′ 0 onjugated to π 1 e . Let ρ σ : π 1 M → Isom( H 3 R ) b e a holonom y represen tation orresp onding to σ ∈ K . By assumption, Γ = ρ σ ( π 1 M ) is a (partiular) w eb group (see for instane [AM℄). More preisely , for all n ∈ N , ρ σ (Γ n ) is a quasifu hsian subgroup of Γ stabilizing a onneted, simply onneted omp onen t Ω n,σ of the domain of dison tin uit y of ρ σ π 1 M , su h that Ω n,σ and Ω m,σ ha v e disjoin t losures if n 6 = m , and that ∂ Ω n,σ on tains no parab oli xed p oin ts of Γ . Let ( H k ,σ ) k ∈ N b e a maximal family of horoballs with pairwise disjoin t in teriors su h that H k ,σ is ρ σ (Γ ′ k ) -in v arian t (su h a family is unique if M has only one torus omp onen t). T o mak e it anonial o v er G F ( M ) , w e ma y x an ordering e 1 = e, e 2 , . . . , e m of the torus omp onen ts of ∂ M , and tak e b y indution H k ,σ , for the k 's in N su h that Γ ′ k is onjugated to π 1 ( e i ) , to b e equiv arian t and maximal with resp et to ha ving pairwise disjoin t in teriors as w ell as ha ving their in terior disjoin t with the in terior of H k ∗ ,σ , for the k ∗ 's in N su h that Γ ′ k ∗ is onjugated to π 1 ( e j ) with j < i . Note that the H k ,σ 's, b esides the ones su h that Γ ′ k is onjugated to π 1 e , are not the maximal horospheres that allo w to dene the heigh t funtions, but this  hanges their v alues only b y a onstan t (uniform on K ). Hene, as K is ompat, there exists δ > 0 su h that for ev ery σ ∈ K , the 1 -on v ex subsets N 1 (Con v Ω n,σ ) and H k ,σ for n, k ∈ N meet pairwise with diameter at most δ . The laim follo ws as in Corollary 5.11 b y applying Corollary 4.11 to X = H 3 R , ǫ 0 = 1 , δ 0 = δ , κ 0 = c ′ 1 ( ∞ ) , C 0 = H 0 ,σ , f 0 = ph C 0 , h ′ 0 = h 0 ( ǫ 0 , δ 0 , κ 0 ) , C 2 n +1 = N 1 (Con v Ω n,σ ) , C 2 n = H n,σ to get a geo desi line e γ in X with presrib ed p enetration in C 0 , and p enetration b ounded b y a onstan t in C n for n 6 = 0 . The niteness of the in tersetion lengths ℓ C 2 n +1 ( e γ ) for n ∈ N implies that e γ sta ys in the on v ex h ull of the limit set of Γ .  Remark. The fat that a lo ally geo desi line sta ys in the on v ex ore of the manifold and do es not on v erge (either w a y) to a usp is equiv alen t with the lo ally geo desi line b eing t w o-sided reurren t. 66 5.4 Presribing the asymptoti p enetration b eha vior Let X b e a prop er geo desi CA T( − 1) metri spae and let ξ ∈ X ∪ ∂ ∞ X . Let ǫ ∈ R ∗ + ∪ { + ∞} , δ , κ ≥ 0 . Let ( C α ) α ∈ A b e a family of ǫ -on v ex subsets of X whi h satises the Almost disjoin tness ondition ( iii ) with parameter δ . F or ea h α ∈ A , let f α b e a κ -p enetration map. Let γ b e a geo desi ra y or line, with 0 in the domain of denition of γ (as w e are only in terested in the asymp oti b eha vior, the  hoie of time 0 is unimp ortan t). These assumptions guaran tee that the set E γ of times t ≥ 0 su h that γ en ters in some C α at time t with f α ( γ ) > δ + κ is disrete in [0 , + ∞ [ , and that α = α t is then unique. The set E γ is nite if f β ( γ ) = + ∞ for some β . Hene E γ = ( t i ) i ∈ N for some initial segmen t N in N , with t i < t i +1 for i, i + 1 in N . With a i ( γ ) = f α t i ( γ ) , the (nite or innite) sequene  a i ( γ )  n ∈ N will b e alled the (nonnegativ e) p enetr ation se quen e of γ with resp et to ( C α , f α ) α ∈ A . In this setion, w e study the asymptoti b eha vior of these p enetration sequenes. W e will only state some results when the C α 's are balls or horoballs, but similar ones are v alid, for instane for ǫ -neigh b orho o ds of geo desi lines in X (see for instane [HPP ℄. W e ma y also presrib e the asymptoti p enetration in one usp, while k eeping the heigh ts in the other usps (uniformly) b ounded. In the follo wing results, w e sho w ho w to presrib e the asymptoti b eha viour of the p en- etration sequene of a geo desi ra y or line with resp et to horoballs and their p enetration heigh t funtions. First, w e pro v e a general result, and w e giv e the more expliit result for Riemannian manifolds as Corollary 5.14. Theorem 5.13 L et X b e a pr op er ge o desi CA T( − 1) metri sp a e, with ∂ ∞ X innite. L et ( H α ) α ∈ A b e a family of hor ob al ls with p airwise disjoint interiors. Assume that ther e exists K ∈ [0 , + ∞ [ and a dense subset Y in ∂ ∞ X suh that, for every ge o desi r ay γ in X with γ (+ ∞ ) ∈ Y , we have lim inf t → + ∞ d ( γ ( t ) , S α ∈ A H α ) ≤ K . L et ξ ∈ X ∪ ∂ ∞ X and c, c ′ ≥ 0 . Assume that for every h ≥ c and α ∈ A suh that ξ / ∈ H α ∪ H α [ ∞ ] , ther e exists a ge o desi r ay or line γ starting fr om ξ and entering H α at time t = 0 with ph H α ( γ ) = h , and with ph H β ( γ ) ≤ c ′ for every β in A − { α } suh that γ (]0 , + ∞ [) me ets H β . L et  a i ( γ ′ )  n ∈ N b e the p enetr ation se quen e of a ge o desi r ay or line γ ′ with r esp e t to ( H α , ph H α ) α ∈ A . Then, for every h ≥ h ∗ = max  c , c ′ + 3 c ′ 1 ( ∞ ) + 10 − 5  , ther e exists a ge o desi r ay or line γ starting fr om ξ suh that lim sup i → + ∞ a i ( γ ) = h . Pro of. T o simplify notation, let f α = ph H α , c ∗ = c ′ + 3 c ′ 1 ( ∞ ) + 10 − 5 , so that h ∗ = max { c ∗ , c } . If a geo desi ra y or line γ starting from ξ meets H α , let t − α ( γ ) and t + α ( γ ) b e the en trane and exit times. Let h ≥ h ∗ , and let α 0 ∈ A su h that ξ / ∈ H α 0 ∪ H α 0 [ ∞ ] , whi h exists b y the assump- tions. As h ≥ h ∗ ≥ c , there exists a geo desi ra y or line γ 0 starting from ξ , en tering H α 0 at time 0 , su h that f α 0 ( γ 0 ) = h , and f α ( γ 0 ) ≤ c ′ for ev ery α 6 = α 0 su h that γ 0 (]0 , + ∞ [) meets H α . W e onstrut, b y indution, sequenes ( γ k ) k ∈ N of geo desi ra ys or lines starting from ξ , ( α k ) k ∈ N of elemen ts of A , and ( t k ) k ∈ N −{ 0 } of elemen ts in [0 , + ∞ [ on v erging to + ∞ , su h that for ev ery k ∈ N , 67 (1) γ k en ters the in terior of H α 0 at time 0 , with d ( γ k (0) , γ k − 1 (0)) ≤ 1 2 k if k ≥ 1 ; (2) γ k en ters H α k and f α k ( γ k ) = h ; (3) if 0 ≤ j ≤ k − 1 , then γ k (]0 , + ∞ [) en ters the in terior of H α j b efore en tering H α k with t − α j ( γ k ) < t k = t + α k − 1 ( γ k ) ; (4) if k ≥ 1 , then for ev ery α su h that γ k (]0 , + ∞ [) meets H α , w e ha v e   f α ( γ k ) − f α ( γ k − 1 )   < 1 2 k if t − α ( γ k ) < t k , and f α ( γ k ) ≤ c ∗ if t k ≤ t − α ( γ k ) < t − α k ( γ k ) , and f α ( γ k ) ≤ c ′ if t − α ( γ k ) ≥ t + α k ( γ k ) . Let us rst pro v e that the existene of su h sequenes implies Theorem 5.13 . By the assertion (1), as γ k (0) sta ys at b ounded distane from γ 0 (0) , up to extrating a subsequene, the sequene ( γ k ) k ∈ N on v erges to a geo desi ra y or line γ ∞ starting from ξ , en tering in H α 0 at time t = 0 , b y on tin uit y of the en tering p oin t in an ǫ -on v ex subset. Let us pro v e that lim sup i → + ∞ a i ( γ ∞ ) = h . The lo w er b ound lim sup i → + ∞ a i ( γ ∞ ) ≥ h is immediate b y a semion tin uit y argumen t. Indeed, for ev ery k > i in N , w e ha v e b y the assertions (2), (3) and (4),   f α i ( γ k ) − h   =   f α i ( γ k ) − f α i ( γ i )   ≤ k − 1 X j = i   f α i ( γ j +1 ) − f α i ( γ j )   ≤ k − 1 X j = i 1 2 j +1 ≤ 1 2 i . Hene b y on tin uit y of f α i , w e ha v e the inequalit y f α i ( γ ∞ ) ≥ h − 1 2 i , whose righ t side on v erges to h as i tends to + ∞ , whi h pro v es the lo w er b ound, as h > c ′ 1 ( ∞ ) and f α i is a c ′ 1 ( ∞ ) -p enetration map in H α i (see Setion 3.1 ). T o pro v e the upp er b ound, assume b y absurd that there exists ǫ > 0 su h that for ev ery λ > 0 , there exists α = α ( λ ) ∈ A su h that γ ∞ en ters H α with f α ( γ ∞ ) ≥ h + ǫ and t − α ( γ ∞ ) > λ + 2 c ′ 1 ( ∞ ) . T ak e λ 0 = max  t i +1 : 1 2 i ≥ ǫ 2  , and α = α ( λ 0 ) By on tin uit y of f α , if k is big enough, w e ha v e f α ( γ k ) ≥ h + ǫ 2 ≥ h ∗ ≥ c ∗ ≥ c ′ . Th us, γ k meets H α as h ∗ > 0 . The en try time is p ositiv e, as d ( γ k (0) , γ ∞ (0)) ≤ c ′ 1 ( ∞ ) and the en trane p oin ts of γ k and γ ∞ in H α are at distane at most c ′ 1 ( ∞ ) , b oth b y Lemma 2.5 , and as the en trane time of γ ∞ in H α is stritly bigger than 2 c ′ 1 ( ∞ ) . Hene, b y the assertion (4), w e ha v e t − α ( γ k ) < t k . Let i ≤ k − 1 b e the minim um elemen t of N su h that for j = i, . . . , k − 1 , the geo desi γ j +1 meets H α at a p ositiv e time with t − α ( γ j +1 ) < t j +1 . By the triangular inequalit y , w e ha v e   t − α ( γ i +1 ) − t − α ( γ ∞ )   ≤ d  γ i +1 ( t − α ( γ i +1 )) , γ ∞ ( t − α ( γ ∞ ))  + d  γ i +1 (0) , γ ∞ (0)  ≤ 2 c ′ 1 ( ∞ ) . Hene t i +1 > t − α ( γ i +1 ) ≥ t − α ( γ ∞ ) − 2 c ′ 1 ( ∞ ) > λ 0 + 2 c ′ 1 ( ∞ ) − 2 c ′ 1 ( ∞ ) = λ 0 . By the denition of λ 0 , w e hene ha v e 1 2 i < ǫ 2 . By the denition of i and b y the assertion (4), w e ha v e f α ( γ i ) = f α ( γ k ) + k − 1 X j = i  f α ( γ j ) − f α ( γ j +1 )  ≥ h + ǫ 2 − k − 1 X j = i 1 2 j +1 ≥ h + ǫ 2 − 1 2 i ≥ h ≥ h ∗ , 68 and in partiular b y the same argumen t as for γ k ab o v e, γ i en ters H α at a p ositiv e time and t − α ( γ i ) < t i . This on tradits the minimalit y of i . This ompletes the pro of, assuming the existene of a sequene with prop erties (1)(4). ξ H α k − 1 γ k ( t k ) γ k (0) H α 0 γ k − 1 (0) γ k − 1 γ k γ ′ k − 1 ( s k ) v γ ′ k − 1 p k H α H α k γ k ( t − α k ( γ k )) γ k − 1 ( t + α k − 1 ( γ k − 1 )) γ k − 1 ( t + α k − 1 ( γ k − 1 ) + A ) Let us no w onstrut the sequenes ( γ k ) k ∈ N , ( α k ) k ∈ N , ( t k ) k ∈ N −{ 0 } . W e ha v e dened γ 0 , α 0 , and they satisfy the prop erties (1)(4). Let k ≥ 1 , and assume that γ k − 1 , α k − 1 , as w ell as t k − 1 if k ≥ 2 , ha v e b een onstruted. As Y is dense in ∂ ∞ X , for ev ery A > 0 , there exists a geo desi ra y or line γ ′ k − 1 starting from ξ with γ ′ k − 1 (+ ∞ ) ∈ Y , en tering in H α 0 at time t = 0 , whi h is v ery lose to γ k − 1 on [0 , t + α k − 1 ( γ k − 1 ) + A ] . By the denition of K , let s k b e the rst time t ≥ t + α k − 1 ( γ k − 1 ) + A su h that there exists α in A with d ( γ ′ k − 1 ( t ) , H α ) ≤ K + 1 , and let α k b e su h an α with d ( γ ′ k − 1 ( s k ) , H α ) minim um. Let p k b e the losest p oin t of H α k to γ ′ k − 1 ( s k ) . Note that ξ / ∈ H α k ∪ H α k [ ∞ ] , if A is big enough (in partiular ompared to K ), as H α 0 and H α k ha v e disjoin t in teriors. By the h yp othesis, let γ k b e a geo desi ra y or line starting from ξ with f α k ( γ k ) = h (whi h pro v es the assertion (2) at rank k as h > 0 ) and f α ( γ k ) ≤ c ′ for ev ery α su h that γ k ([ t + α k ( γ k ) , + ∞ [) en ters H α . As a CA T( − 1) metri spae is log(1 + √ 2) - h yp erb oli, the geo desi ] ξ , p k ] is on tained in the log(1 + √ 2) - neigh b ourho o d of the union ] ξ , γ ′ k − 1 ( s k )] ∪ [ γ ′ k − 1 ( s k ) , p k ] . By Lemma 2.5 , w e ha v e d ( γ k ( t − α k ( γ k )) , ] ξ , p k ]) ≤ c ′ 1 ( ∞ ) , and therefore ] ξ , γ k ( t − α k ( γ k ))] is on tained in the ( c ′ 1 ( ∞ ) + log(1 + √ 2)) -neigh b ourho o d of ] ξ , γ ′ k − 1 ( s k )] ∪ [ γ ′ k − 1 ( s k ) , p k ] . Up to  ho osing A big enough, w e ma y hene assume that γ k is v ery lose to γ k − 1 b et w een the times 0 and t + α k − 1 ( γ k − 1 ) + 1 . Using this and prop erties (1) and (3) at rank k − 1 , w e ha v e • γ k do es en ter the in terior of H α 0 , at a time that w e ma y assume to b e 0 , with d ( γ k (0) , γ k − 1 (0)) ≤ 1 2 k (this pro v es the assertion (1) at rank k ); • for 0 ≤ j ≤ k − 1 , as γ k − 1 passes in the in terior of H α j at a time stritly b et w een 0 and t + α k − 1 ( γ k − 1 ) , b y the indutiv e assertions (3) if k 6 = 1 and j ≤ k − 2 , or (1) if k = 1 or (2) if j = k − 1 , so do es the geo desi ra y or line γ k ; this allo ws, in partiular, to dene t k = t + α k − 1 ( γ k ) , and pro v es the assertion (3) at rank k ; • for ev ery α su h that γ k (]0 , + ∞ [) meets H α and t − α ( γ k ) < t k , w e ma y assume b y on tin uit y that   f α ( γ k ) − f α ( γ k − 1 )   < 1 2 k . Hene, to pro v e the assertion (4) at rank k , w e onsider α ∈ A su h that γ k meets H α with t k ≤ t − α ( γ k ) < t − α k ( γ k ) , and w e pro v e that f α ( γ k ) ≤ c ∗ . W e ma y assume that f α ( γ k ) > 0 . Let v b e the highest p oin t of γ k in H α , whi h, b y disjoin tness, b elongs to 69 ] γ k ( t − α ( γ k )) , γ k ( t − α k ( γ k ))[ . Let u b e a p oin t in ] ξ , γ ′ k − 1 ( s k )] ∪ [ γ ′ k − 1 ( s k ) , p k ] at distane at most c ′ 1 ( ∞ ) + log (1 + √ 2) from v . Assume rst that u ∈ [ γ ′ k − 1 ( s k ) , p k ] . Note that b y the minimalit y assumption on α k , the p oin t u then do es not b elong to H α . As c ∗ ≥ 2 c ′ 1 ( ∞ ) + 2 log (1 + √ 2) = 3 c ′ 1 ( ∞ ) , this implies that f α ( γ k ) ≤ 2 d ( u, v ) ≤ c ∗ . Assume no w that u = γ ′ k − 1 ( t ) with t ∈ [ t + α k − 1 ( γ k − 1 ) + A, s k [ . Then b y the minimalit y of s k , the p oin t u again do es not b elong to H α (it is in fat at distane at least K + 1 from H α ). Hene similarly f α ( γ k ) ≤ c ∗ . Finally , assume that u = γ ′ k − 1 ( t ) with t ∈ [0 , t + α k − 1 ( γ k − 1 ) + A [ . Let u ′ b e a p oin t on γ k − 1 ([ t + α k − 1 ( γ k − 1 ) , t + α k − 1 ( γ k − 1 ) + A ]) with d ( u, u ′ ) ≤ 10 − 5 / 2 (as γ k − 1 and γ ′ k − 1 w ere assumed to b e v ery lose on that range). The p oin t u ′ is at distane at most c ′ / 2 from a p oin t in the omplemen t of H α , as if it b elongs to the in terior of H α , then f α ( γ k − 1 ) ≤ c ′ b y the indutiv e h yp othesis (4) on γ k − 1 . Hene f α ( γ k ) ≤ 2 d ( v , u ′ ) + 2 ( c ′ / 2) ≤ 2 c ′ 1 ( ∞ ) + 10 − 5 = c ∗ . This pro v es the result.  Remark. The pro of when ( H α ) α ∈ A is a family of balls of radius R > 0 , replaing c ′ 1 ( ∞ ) b y c ′ 1 ( R ) , and assuming b oth in the h yp othesis and in the onlusion that h ≤ c ′′ for some c ′′ , is the same. Corollary 5.14 L et X b e a  omplete simply  onne te d R iemannian manifold with se - tional urvatur e at most − 1 and dimension at le ast 3 , and let  H α  α ∈ A b e a family of hor ob al ls in X with disjoint interiors. Assume that ther e exists K ∈ [0 , + ∞ [ and a dense subset Y in ∂ ∞ X suh that, for every ge o desi r ay γ in X with γ (+ ∞ ) ∈ Y , we have lim inf t → + ∞ d ( γ ( t ) , S α ∈ A H α ) ≤ K . Then, for every ξ ∈ X ∪ ∂ ∞ X and h ≥ c ′′ 1 ( ∞ , 0 , 0) + 4 c ′ 1 ( ∞ ) + 10 − 5 ≈ 13 . 5542 , ther e exists a ge o desi r ay or line γ starting fr om ξ suh that, with ( a i ( γ )) n ∈ N the p ene- tr ation se quen e of γ with r esp e t to ( H α , ph H α ) α ∈ A , we have lim sup i → + ∞ a i ( γ ) = h . Pro of. Let c = c ′′ 1 ( ∞ , 0 , 0) , c ′ = c ′′ 1 ( ∞ , 0 , 0) + c ′ 1 ( ∞ ) . W e apply Theorem 5.1 with ǫ = ∞ , δ = 0 , κ = c ′ 1 ( ∞ ) , ξ 0 = ξ , C 0 = H α where α ∈ A satises ξ / ∈ H α ∪ H α [ ∞ ] , f 0 = ph C 0 , ( C n ) n ≥ 1 is ( H β ) β ∈ A −{ α } (up to indexation). Then the assumptions of Theorem 5.13 are satised. An easy omputation of h ∗ in Theorem 5.13 then yields the result.  Remark. Using Theorem 5.2 instead of Theorem 5.1, the same statemen t when ( H α ) α ∈ A is a family of balls of radius R > 0 , for h ∈ [ c ′′ 1 ( R, 0 , 0) + 4 c ′ 1 ( R ) + 10 − 5 , 2 R − c ′ 1 ( R )] holds true. As in Setion 5.1, w e onsider a omplete, nonelemen tary , geometrially nite Rieman- nian manifold M , and e an end of M . The asymptoti height sp e trum of the pair ( M , e ) is LimsupS p( M , e ) =  lim sup t →∞ h t e ( γ ( t )) : γ ∈ T 1 M  . In lassial Diophan tine appro ximation, the L agr ange sp e trum is the subset of [0 , + ∞ [ onsisting of the appr oximation  onstants c ( x ) of an irrational real n um b er x b y rational n um b ers p/q , dened b y c ( x ) = − 2 log µ = lim inf q →∞ | q | 2   x − p q   . 70 Using the w ell kno wn onnetion b et w een the Diophan tine appro ximation of real n um b ers b y rational n um b ers and the ation of the mo dular group PSL 2 ( Z ) on the upp er halfplane mo del of the real h yp erb oli plane, the asymptoti heigh t sp etrum of the mo dular orbifold PSL 2 ( Z ) \ H 2 R is the image of the Lagrange sp etrum b y the map t 7→ − 2 log t (see for instane [HP3 , Theo. 3.4℄). Hall [Hal1, Hal2℄ sho w ed that the Lagrange sp etrum on tains an in terv al [0 , c ] for some c > 0 . The maximal su h in terv al [0 , µ ] (whi h is losed as the Lagrange sp etrum is losed, b y Cusi k's result, see for instane [CF ℄), alled Hal l's r ay , w as determined b y F reiman [F re ℄ (see also [Slo ℄ where the map t 7→ 1 /t has to b e applied). The geometri in terpretation of F reiman's result in our on text is that LimsupS p(PSL 2 ( Z ) \ H 2 R ) on tains the maximal in terv al [ c, + ∞ ] with c = − 2 log  49199 3569 22215 64096 + 28 3748 √ 462  ≈ 3 . 0205 . The follo wing result is the asymptoti analog of Corollary 5.4, and has a ompletely similar pro of. Theorem 1.6 in the in tro dution follo ws, sine ( c ′′ 1 ( ∞ , 0 , 0) + 4 c ′ 1 ( ∞ ) + 10 − 5 ) / 2 ≈ 6 . 7771 . The result pro v es the existene of Hall's ra y in our geometri on text, whi h is m u h more general (there is no assumption of arithmeti nature, nor of onstan t urv ature nature), and with a univ ersal onstan t (though w e do not kno w the optimal one) 6 . 7771 whi h is not to o far from the geometri F reiman onstan t 3 . 0205 from the ab o v e partiular ase. Corollary 5.15 L et M b e a  omplete, nonelementary, ge ometri al ly nite R iemannian manifold with se tional urvatur e at most − 1 and dimension at le ast 3 , and let e b e a usp of M . Then LimsupS p( M , e )  ontains the interval [( c ′′ 1 ( ∞ , 0 , 0) + 4 c ′ 1 ( ∞ ) + 10 − 5 ) / 2 , + ∞ ] .  In the next setion 6 , w e onsider a n um b er of arithmetially dened examples, illustrat- ing this last result. But w e need rst to reall some prop erties and do some omputations in the real and omplex h yp erb oli spaes. 6 Appliations to Diophan tine appro ximation in negativ ely urv ed manifolds 6.1 On omplex h yp erb oli geometry and the Heisen b erg group T o failitate omputations, w e iden tify elemen ts in C n − 1 with their o ordinate olumn matries. W e will denote b y A ∗ = t A the adjoin t matrix of a omplex matrix A . In partiular, the standard hermitian salar pro dut of w, w ′ ∈ C n − 1 is w ∗ w ′ = P n − 1 i =1 w i w ′ i . W e also use the notation | w | 2 = w ∗ w . Let H n C b e the Siegel domain mo del of omplex h yp erb oli n -spae whose underlying set is H n C = { ( w 0 , w ) ∈ C × C n − 1 : 2 Re w 0 − | w | 2 > 0 } , and whose Riemannian metri is ds 2 C = 4 (2 Re w 0 − | w | 2 ) 2  ( dw 0 − dw ∗ w )(( dw 0 − w ∗ dw ) + (2 Re w 0 − | w | 2 ) dw ∗ dw  , 71 (see for instane [Gol, Set. 4.1℄). Complex h yp erb oli spae has onstan t holomorphi setional urv ature − 1 , hene its real setional urv atures are b ounded b et w een − 1 and − 1 4 . Its b oundary at innit y is ∂ ∞ H n C = { ( w 0 , w ) ∈ C × C n − 1 : 2 Re w 0 − | w | 2 = 0 } ∪ {∞} . The horoballs en tered at ∞ in H n C are the subsets H s = { ( w 0 , w ) ∈ C × C n − 1 : 2 Re w 0 − | w | 2 ≥ s } , for s > 0 . Note that the subset H 1 C = { ( w 0 , w ) ∈ H n C : w = 0 } is the righ t halfplane mo del of the real h yp erb oli plane with onstan t urv ature − 1 , and it is totally geo desi in H n C . In partiular, the (unit sp eed) geo desi line starting from ∞ , ending at (0 , 0) ∈ ∂ ∞ H n C and meeting the horosphere ∂ H 2 at time t = 0 is the map c 0 : R → H n C dened b y c 0 : t 7→ ( e − t , 0) . Let q b e the nondegenerate Hermitian form − z 0 z n − z n z 0 + | z | 2 of signature (1 , n ) on C × C n − 1 × C with o ordinates ( z 0 , z , z n ) . This is not the form onsidered in [Gol , page 67℄, hene w e need to do some omputations with it, but it is b etter suited for our purp oses. The Siegel domain H n C em b eds in the omplex pro jetiv e n -spae P n ( C ) b y the map (using homogeneous o ordinates) ( w 0 , w ) 7→ [ w 0 : w : 1] . Its image is the negativ e one of q , that is { [ z 0 : z : z n ] ∈ P n ( C ) : q ( z 0 , z , z n ) < 0 } . This em b edding extends on tin uously to the b oundary at innit y , b y mapping ( w 0 , w ) ∈ ∂ ∞ H n C − {∞} to [ w 0 : w : 1] and ∞ to [1 : 0 : 0] , so that the image of ∂ ∞ H n C is the n ull one of q , that is { [ z 0 : z : z n ] ∈ P n ( C ) : q ( z 0 , z , z n ) = 0 } . W e use matries b y blo  ks in the deomp osition C × C n − 1 × C . Let Q =   0 0 − 1 0 I 0 − 1 0 0   b e the matrix of q . If X =   a γ ∗ b α A β c δ ∗ d   , then Q − 1 X ∗ Q =   d − β ∗ b − δ A ∗ − γ c − α ∗ a   . If U Q is the group of in v ertible matries with omplex o eien ts preserving the hermitian form q , then X b elongs to U Q if and only if X is in v ertible with in v erse Q − 1 X ∗ Q . In partiular, if X b elongs to U Q , then                c d − δ ∗ δ + dc = 0 ab − γ ∗ γ + ba = 0 − αβ ∗ + AA ∗ − β α ∗ = I c b − δ ∗ γ + da = 1 dα − Aδ + cβ = 0 bα − Aγ + aβ = 0 . (- 33 -) 72 The group U Q ats pro jetiv ely on P n ( C ) , preserving the negativ e one of q , hene it ats on H n C . W e will denote in the same w a y the ation of U Q on H n C and the ation of U Q on the image of H n C in P n ( C ) . It is w ell kno wn (see for instane [Gol ℄) that U Q preserv es the Riemannian metri of H n C . The Heisen b erg group Heis 2 n − 1 is the real Lie group with underlying spae C n − 1 × R and la w ( ζ , v )( ζ ′ , v ′ ) = ( ζ + ζ ′ , v + v ′ − 2 Im ζ ∗ ζ ′ ) . It has a Lie group em b edding in U Q , dened b y ( ζ , v ) 7→ u ζ , v =   1 ζ ∗ | ζ | 2 2 − i v 2 0 I ζ 0 0 1   , whose image preserv es the p oin t ∞ as w ell as ea h horoball en tered at ∞ , as an easy omputation sho ws. The Cygan distan e (see [Gol, page 160℄) on Heis 2 n − 1 is the unique left-in v arian t dis- tane d Cyg su h that d Cyg ((0 , 0) , ( ζ , v )) = ( | ζ | 4 + v 2 ) 1 / 4 . W e in tro due the mo die d Cygan distan e d ′ Cyg as the unique left-in v arian t distane d ′ Cyg su h that d ′ Cyg ((0 , 0) , ( ζ , v )) = (( | ζ | 4 + v 2 ) 1 / 2 + | ζ | 2 ) 1 / 2 . It is straigh tforw ard to  he k that d ′ Cyg is indeed a distane, in the same w a y as the Cygan distane, see for instane [KR℄, and that it is equiv alen t to the Cygan distane, d Cyg ≤ d ′ Cyg ≤ √ 2 d Cyg . Hene, its indued length distane is equiv alen t to the Carnot-Carathéodory distane on the Heisen b erg group Heis 2 n − 1 (see [Gol, page 161℄). As the ation of Heis 2 n − 1 on ∂ ∞ H n C − {∞} is simply transitiv e, d Cyg and d ′ Cyg dene distanes on ∂ ∞ H n C − {∞} , whi h are in v arian t under the ation of Heis 2 n − 1 . W e also all these distanes the Cygan distan e and the mo die d Cygan distan e , and again denote them b y d Cyg and d ′ Cyg . Expliitly , these distanes are giv en b y d Cyg ( u ζ , v (0 , 0) , u ζ ′ ,v ′ (0 , 0)) = d Cyg (( ζ , v ) , ( ζ ′ , v ′ )) , and the similar expression for the mo died Cygan distane. Lemma 6.1 The distan e d Cyg (r esp. d ′ Cyg ) is the unique distan e on ∂ ∞ H n C − {∞} invari- ant under Heis 2 n − 1 suh that d ′ Cyg (( w 0 , w ) , (0 , 0)) = p 2 | w 0 | (r esp. d ′ Cyg (( w 0 , w ) , (0 , 0)) = p 2 | w 0 | + | w | 2 ). Pro of. F or ev ery ( w 0 , w ) in ∂ ∞ H n C − {∞} , note that ( w 0 , w ) = u ζ , v (0 , 0) if and only if v = − 2 Im w 0 and ζ = w , and that 2 Re w 0 = | w | 2 . Hene d ′ Cyg ( u ζ , v (0 , 0) , (0 , 0)) = ((4 R e 2 w 0 + 4 Im 2 w 0 ) 1 / 2 + | w | 2 ) 1 / 2 = p 2 | w 0 | + | w | 2 . A similar pro of giv es the result for the Cygan distane.  73 In partiular, if n = 2 , then d ′ Cyg is indeed dened as in the statemen t of Theorem 1.8 in the in tro dution. Let d H n C b e the Riemannian distane on H n C , and d ′ H n C = 1 2 d H n C b e the Riemannian distane of the Riemannian metri of H n C renormalized to ha v e maximal real setional urv atures − 1 . Prop osition 6.2 F or every ξ , ξ ′ in ∂ ∞ H n C − {∞} , for every s 0 > 0 , the distan e ℓ ′ for the r enormalize d R iemannian distan e d ′ H n C b etwe en the hor ob al l H s 0 and the hor ob al l  enter e d at ξ and tangent to the ge o desi line b etwe en ∞ and ξ ′ is, if these hor ob al ls ar e disjoint, ℓ ′ = − log d ′ Cyg ( ξ , ξ ′ ) + 1 2 log( s 0 2 ) . Pro of. By in v ariane of the mo died Cygan distane, of ea h horoball en tered at ∞ , and of the normalized Riemannian distane, b y the ation of the Heisen b erg group, w e ma y assume that ξ = (0 , 0) . Let ( ζ , v ) ∈ Heis 2 n − 1 su h that ξ ′ = u ζ , v ( ξ ) . As u ζ , v sends geo desi lines to geo desi lines, and xes ∞ , the geo desi line (for d H n C ) starting from ∞ and ending at ξ ′ is u ζ , v ◦ c 0 , whi h b y an easy omputation is u ζ , v ◦ c 0 : t 7→ ( e − t + ( | ζ | 2 − iv ) / 2 , ζ ) . The matrix X 0 =   0 0 1 0 I 0 1 0 0   b elongs to U Q , as X − 1 0 = X 0 and Q − 1 X ∗ 0 Q = X 0 , and the orresp onding isometry of H n C sends ∞ ∈ ∂ ∞ H n C to (0 , 0) ∈ ∂ ∞ H n C . Hene X 0 sends the horoballs en tered at ∞ to the horoballs en tered at (0 , 0) . Let s > 0 , an easy omputation sho ws that X 0 H s = { ( w 0 , w ) ∈ C × C n − 1 : 2 Re w 0 − | w | 2 ≥ s | w 0 | 2 } . F or ev ery t in R , the p oin t u ζ , v ◦ c 0 ( t ) b elongs to the horosphere X 0 ∂ H s if and only if 2 Re ( e − t + ( | ζ | 2 − iv ) / 2) − | ζ | 2 = s | e − t + ( | ζ | 2 − iv ) / 2 | 2 , that is, if and only if s e − 2 t + ( s | ζ | 2 − 2) e − t + s 4 ( | ζ | 4 + v 2 ) = 0 . The horoball X 0 H s is hene tangen t to the geo desi line u ζ , v ◦ c 0 if and only if the ab o v e quadrati equation with unkno wn e − t has a double solution, that is, if and only if its disriminan t ∆ is 0 . An easy omputation giv es − ∆ = s 2 v 2 + 4 s | ζ | 2 − 4 . Th us, the horoball X 0 H s is tangen t to u ζ , v ◦ c 0 if and only if s = 2 p | ζ | 4 + v 2 + | ζ | 2 . (- 34 -) As the geo desi line c 0 passes through the p oin t at innit y of b oth horoballs H s 0 and X 0 H s (whi h ha v e disjoin t in teriors if s 0 is big enough), the Riemannian distane b et w een them is the length of the subsegmen t of c 0 joining them. Note that c 0 meets X 0 ∂ H s at ( 2 s , 0) . Hene, b y an easy omputation in H 1 C , ℓ ′ = d ′ H n C ( H s 0 , X 0 H s ) = 1 2 d H n C ( H s 0 , X 0 H s ) = 1 2 d H n C (( s 0 2 , 0) , ( 2 s , 0)) = 1 2 (log s 0 2 − log 2 s ) . 74 By Equation (- 34 - ), the result follo ws.  F or ev ery X in U Q , w e will denote b y c = c ( X ) its (3 , 1) -o eien t in its matrix b y blo  ks. Note that X xes ∞ if and only if c = 0 , b y the equations (- 33 - ). Equiv alen tly , b y the same set of equations, a matrix xes ∞ if and only if it is upp er triangular (this is the main reason wh y w e  hose the hermitian form q rather than the one in [ Gol℄). The follo wing lemma is ompletely analogous to Prop osition 5.14 of [ HP4 ℄, but as w e are using a dieren t quadrati form, w e need to giv e a pro of. Lemma 6.3 F or every X in U Q and every s > 0 suh that the hor ob al ls H s and X H s have disjoint interiors, we have d ′ ( H s , X H s ) = log | c | + log s 2 . Pro of. As H s and X H s ha v e disjoin t in teriors, X do es not x ∞ , hene c 6 = 0 . Left and righ t m ultipliation of X b y an elemen t u ζ , v for some ( ζ , v ) in Heis 2 n − 1 do es not  hange the o eien t c of X , nor do es it  hange d ′ ( H s , X H s ) = 1 2 d ( H s , X H s ) , as u ζ , v preserv es the distane d and ea h horosphere en tered at ∞ . Hene, as Heis 2 n − 1 ats transitiv ely on ∂ ∞ H n C − {∞} , w e ma y assume that X ∞ = (0 , 0) and that X − 1 ∞ = (0 , 0) . As X ∞ = (0 , 0) , the o eien ts a, α of X are 0 , and hene b y the seond equation of ( - 33 - ), the o eien t γ is 0 . As X − 1 ∞ = (0 , 0) , the o eien ts d, δ of X are 0 , and hene b y the fth equation of (- 33 - ), the o eien t β is 0 . Therefore, b y the third and fourth equation of (- 33 - ), the matrix X has the form   0 0 1 c 0 A 0 c 0 0   , with A unitary . An easy omputation, similar to the one w e already did with X 0 , sho ws that X H s = { ( w 0 , w ) ∈ C × C n − 1 : 2 Re w 0 − | w | 2 ≥ s | c | 2 | w 0 | 2 } . Hene, as ab o v e, d ′ ( H s , X H s ) = 1 2 d ( H s , X H s ) = 1 2 d (( s 2 , 0) , ( 2 s | c | 2 , 0)) = log | c | + lo g s 2 .  Let m b e a squarefree p ositiv e in teger, let K − m = Q ( i √ m ) b e the orresp onding imaginary quadrati n um b er eld, and let − m b e the ring of in tegers of K − m . An or der in K − m is a unitary subring of − m whi h is a free Z -mo dule of rank 2 . W e use for instane [Co x ,  hap. 7℄ for a general referene on these ob jets. An example of an order in K − m is Z [ i √ m ] , and − m is the maximal order of K − m . In partiular, on tains a Q -basis of K − m , and the eld of frations of is K − m . Let ω b e the elemen t of with Im ω > 0 su h that = Z [ ω ] = Z + ω Z . As is stable b y omplex onjugation, the subset SU Q () = SU Q ∩ M n +1 () is a disrete subgroup of the semi-simple onneted real Lie group SU Q = U Q ∩ SL n +1 ( C ) . Let I b e a non-zero ideal of . W e denote b y Γ C , I the preimage, b y the group morphism SU Q () → SL n +1 ( / I ) of redution mo dulo I , of the parab oli subgroup of matries whose rst olumn has all its o eien ts 0 exept the rst one. As / I is nite ( I is nonzero), Γ C , I is a nite index subgroup of SU Q () as 75 Reall that a horoball H en tered at a p oin t ξ in a CA T( − 1) metri spae X is pr e isely invariant under a group of isometries Γ if for ev ery g ∈ Γ that do es not x ξ , the in tersetion g ◦ H ∩ ◦ H is empt y . Lemma 6.4 F or v = 2 Im ω if Re ω ∈ Z , and v = 4 I m ω otherwise, the hor ob al l H v is pr e isely invariant under Γ C , I . F urthermor e, if I = = − 1 , then H 2 is the maximal hor ob al l  enter e d at ∞ whih is pr e isely invariant under Γ C , I . Pro of. With v as in the statemen t, the elemen t − iv / 2 b elongs to , as i Im ω = ω − Re ω b elongs to if Re ω ∈ Z , and 2 i Im ω = ω − ω b elongs to (whi h is stable b y onjugation). Hene u 0 ,v b elongs to Γ C , I . It follo ws for instane from [HP1 , Prop. 5.7℄ (whi h is an easy onsequene of the omplex h yp erb oli Shimizu inequalit y of Kamiy a [Kam℄ and P ark er [P ar ℄) that the horoball H v is preisely in v arian t (the hermitian form q in [HP1 ℄ is not the same one as the ab o v e, but it is equiv alen t b y a p erm utation of o ordinates, hene w e ma y indeed apply [HP1 , Prop. 5.7℄). If I = = − 1 , then X 0 dened ab o v e b elongs to Γ C , I and ω = i . By Lemma 6.3, w e ha v e d ( H 2 , X 0 H 2 ) = 0 , hene the last assertion follo ws.  F or ev ery ( a, α, c ) ∈ × n − 1 × , let h a, α, c i b e the ideal of generated b y a , c and the omp onen ts of α . Prop osition 6.5 If n = 2 and = − m , then (1) for every I , the set of p ar ab oli xe d p oints of Γ C , I is exatly the set of p oints in ∂ ∞ H n C having homo gene ous  o or dinates in P n ( C ) that ar e elements in K − m ; (2) the orbit Γ C , I ·∞ is exatly the set of p oints in ∂ ∞ H n C having homo gene ous  o or dinates in P n ( C ) of the form [ a : α : c ] with ( a, α, c ) ∈ × I n − 1 × I , 2 Re ac = | α | 2 and h a, α, c i = ; (3) if m = 1 , 2 , 3 , 7 , 11 , 19 , 43 , 67 , 163 and I = , then Γ C , I has only one orbit of p ar ab oli xe d p oints. Pro of. (1) If I = , then the rst result is due to Holzapfel [ Hol1 ℄, see [Hol2 , page 280℄. As Γ C , I has nite index in SU Q () , and as a disrete group and a nite index subgroup ha v e the same set of parab oli xed p oin ts, the rst laim follo ws. (2) A result of F eustel [F eu ℄ (see [Hol2, page 280℄, [Zin℄) sa ys that the map whi h asso- iates to a parab oli xed p oin t of SU Q () the frational ideal generated b y its homogeneous o ordinates in − m indues a bijetion from the set of orbits under SU Q () of parab oli xed p oin ts of SU Q () to the set of ideal lasses of K − m . As ∞ orresp onds to [1 : 0 : 0] whose o ordinates generate the trivial frational ideal, the seond laim follo ws if I = , as w ell as laim (3). If M   1 0 0   =   a α c   , then   a α c   is the rst olumn of the matrix M , so that the seond laim if I 6 = follo ws b y the denition of Γ C , I .  76 6.2 Quaternions and 5 -dimensional real h yp erb oli geometry Let H b e Hamilton's quaternion algebra o v er R , generated as a real v etor spae b y the standard basis 1 , i, j, k , with pro duts k = ij = − j i , i 2 = − 1 , j 2 = − 1 and unit 1 . Reall that the  onjugate of the quaternion z = x 1 + x 2 i + x 3 j + x 4 k is z = x 1 − x 2 i − x 3 j − x 4 k , whi h satises z w = w z , and that its absolute value (or square ro ot of its redued norm) is | z | = p N ( z ) = √ z z = √ z z = q x 2 1 + x 2 2 + x 2 3 + x 2 4 . The Dieudonné determinant (see [ Die ℄ and [Asl℄) ∆ is the group morphism from the group GL 2 H of in v ertible 2 × 2 matries with o eien ts in H to R ∗ + , giv en b y ∆(  a b c d  ) =  | ad − aca − 1 b | if a 6 = 0 | cb − cac − 1 d | if c 6 = 0 . W e will denote b y SL 2 ( H ) the group of 2 × 2 quaternioni matries with Dieudonné determi- nan t 1 (this notation is dieren t, hene should not b e onfused with the notation SL(2 , C n ) for n = 3 and C 3 = H of V ahlen and Ahlfors [Ahl℄, see also [MWW ℄, giving a desription of the isometry group of the real h yp erb oli ( n + 1) -spae using the 2 n -dimensional real Cliord algebra C n ). W e refer for instane to [Kel℄ for more information on SL 2 H . The group SL 2 ( H ) ats on the Alexandro v ompatiation H ∪ {∞} of H b y  a b c d  · z =    ( az + b )( cz + d ) − 1 if z 6 = ∞ , − c − 1 d ac − 1 if z = ∞ , c 6 = 0 ∞ otherwise . It is w ell kno wn (see for instane [Kel℄) that PSL 2 ( H ) = SL 2 ( H ) / {± Id } is the orien tation preserving onformal group of the 4 -sphere H ∪ {∞} with its standard onformal struture dened b y the 4 -dimensional Eulidean spae ( H , | · | ) . In the upp er halfspae mo del H 5 R of the 5 -dimensional real h yp erb oli spae with onstan t urv ature − 1 , onsider the o ordinates ( z , t ) with z ∈ H and t > 0 (alled the verti al  o or dinate ), so that ∂ ∞ H 5 R iden ties with the union of H (for t = 0 ) and of {∞} . By the P oinaré extension pro edure (see for instane [Bea , Set. 3.3℄), the group PSL 2 ( H ) hene iden ties to the group of orien tation preserving isometries of H 5 R . W e will denote the Riemannian distane on H 5 R b y d H 5 R . Lemma 6.6 [ Hel , The o. 1.-2)℄ F or every g =  a b c d  in SL 2 ( H ) , and ( z , t ) in H 5 R , the verti al  o or dinate of g ( z , t ) is t | cz + d | 2 + | c | 2 t 2 . Pro of. As [Hel ℄ is an announemen t, w e giv e a pro of for the sak e of ompleteness. The pro of is an adaptation of the pro of for SL 2 ( C ) in [Bea , page 58℄, the main problem onsists of b eing areful with the nonomm utativit y of H . W e ma y assume that c 6 = 0 , as the map z 7→ αz β + γ for α, β , γ in H ∗ × H ∗ × H is an Eulidean similitude of ratio | αβ | . Dene the isometri spher e of g to b e the sphere S g of en ter − c − 1 d and radius 1 | c | in the Eulidean spae ( H , | · | ) . By the denition of an Eulidean reetion with resp et to a sphere in this Eulidean spae, the map σ : z 7→ − c − 1 d + 1 | c | 2 z + c − 1 d | z + c − 1 d | 2 77 is the Eulidean reetion with resp et to the sphere S g . An easy omputation sho ws that the map ϕ = g ◦ σ is z 7→ ( b − ac − 1 d )( z c + d ) + ac − 1 , whi h is an Eulidean isometry , as z 7→ z is, and | cb − cac − 1 d | = 1 . The P oinaré extension of ϕ preserv es the v ertial o ordinates, and the P oinaré extension of σ is the Eulidean reetion with resp et to the sphere in H 5 R whose equator is S g . As g = ϕ ◦ σ , the result follo ws b y an easy omputation.  The horoballs en tered at ∞ in H 5 R are the subsets H s for s > 0 , where H s = { ( z , t ) ∈ H × ]0 , + ∞ [ : t ≥ s } . Lemma 6.7 F or every g =  a b c d  in SL 2 H , and every s > 0 suh that the hor ob al ls H s and g H s have disjoint interiors, we have d ( H s , X H s ) = 2 log | c | + 2 log s . Pro of. As H s and g H s ha v e disjoin t in teriors, w e ha v e c 6 = 0 . The map g sends the geo desi line b et w een − c − 1 d and ∞ to the geo desi line b et w een ∞ and ac − 1 , hene the p oin t ( − c − 1 d, s ) of in tersetion of the rst line with H s to the p oin t g ( − c − 1 d, s ) of in tersetion of the seond line with g H s . The v ertial o ordinate of g ( − c − 1 d, s ) is 1 | c | 2 s b y the previous lemma 6.6. Hene the result follo ws b y an easy omputation of h yp erb oli distanes.  W e will use [Vig℄ and [MR, Setion 2℄ as general referenes on quaternion algebras. Let A ( Q ) b e a quaternion algebra o v er Q , whi h is r amie d o v er R , that is, the real algebra A ( Q ) ⊗ Q R is isomorphi to Hamilton's algebra H . W e iden tify A ( Q ) ⊗ Q R and H b y an y su h isomorphism. Let ′ b e an or der of A ( Q ) , that is an unitary subring whi h is a nitely generated Z -mo dule generating the Q -v etor spae A ( Q ) . F or instane, if A ( Q ) = { x 1 + x 2 i + x 3 j + x 4 k ∈ H : x 1 , x 2 , x 3 , x 4 ∈ Q } , w e an tak e ′ = { x 1 + x 2 i + x 3 j + x 4 k ∈ H : x 1 , x 2 , x 3 , x 4 ∈ Z } , or the Hurwitz ring ′ =  x 1 1 + i + j + k 2 + x 2 i + x 3 j + x 4 k ∈ H : x 1 , x 2 , x 3 , x 4 ∈ Z  , whi h is a maximal order. Let I ′ b e a non-zero t w o-sided ideal in the ring ′ . W e denote b y Γ I ′ the preimage in the group morphism SL 2 ( ′ ) → GL 2 ( ′ / I ′ ) of re- dution mo dulo I ′ of the subgroup of upp er triangular matries. As ′ / I ′ is nite ( I ′ is nonzero), Γ I ′ is a nite index subgroup of SL 2 ( ′ ) . Lemma 6.8 The hor ob al l H 1 is pr e isely invariant under Γ I ′ . F urthermor e, if I ′ = ′ , then H 1 is the maximal hor ob al l  enter e d at ∞ whih is pr e isely invariant under Γ I ′ . Pro of. The elemen t  1 1 0 1  b elongs to Γ I ′ . It follo ws from [Kel, page 1091℄ that the horoball H 1 is preisely in v arian t. If I ′ = ′ , then g =  0 1 − 1 0  b elongs to Γ I ′ , and b y Lemma 6.7 , d ( H 1 , g H 1 ) = 0 , hene the last assertion follo ws.  78 6.3 On arithmeti latties The follo wing result follo ws from the w ork of Borel and Harish-Chandra [BHC℄ and of Borel [Bor2 , Theo. 1.10℄ (see [Bor1℄ for an elemen tary presen tation of semisimple algebrai groups). T w o subgroups A and B of a group C are said to b e  ommensur able in this theorem if A ∩ B has nite index in b oth A and B . Theorem 6.9 [BHC , Bor2 ℄ L et G b e a  onne te d semisimple algebr ai gr oup dene d over Q of R -r ank one and P b e a minimal p ar ab oli sub gr oup of G dene d over Q , let G = G ( R ) 0 and P = G ∩ P ( R ) , let Γ b e a sub gr oup of G  ommensur able to G ( Z ) ∩ G , then Γ is a latti e in G , and the set of p ar ab oli xe d p oints of Γ on G/P is G ( Q ) P .  Su h a subgroup Γ will b e alled an arithmeti latti e in G . Note that the R -rank assumption is equiv alen t to the fat that for ev ery (or equiv alen tly an y) maximal ompat subgroup K of the Lie group G , the asso iated symmetri spae X = G/K ma y b e endo w ed with a G -in v arian t Riemannian metri with setional urv ature at most − 1 . Su h a metri is then unique up to m ultipliation b y a p ositiv e onstan t, and P is the stabilizer of a p oin t in the b oundary at innit y ∂ ∞ X . The orbital map at this p oin t hene indues a G -equiv arian t homeomorphism b et w een G/P and ∂ ∞ X . Note that there is a terminology problem: b y a parab oli elemen t, w e mean an isometry of X ha ving a unique xed p oin t (alled a p ar ab oli xe d p oint ) on X ∪ ∂ ∞ X , that b elongs to ∂ ∞ X , but the set of real p oin ts of a parab oli subgroup of G also on tains non parab oli elemen ts ! Examples. (1) Let m b e a squarefree p ositiv e in teger, and let I b e a non-zero ideal in an order in the ring of in tegers − m of the imaginary quadrati n um b er eld K − m = Q ( i √ m ) . Let (1 , ω ) b e a basis of as a Z -mo dule. It is also a basis of K − m as a Q -v etor spae, and of C as an R -v etor spae. If x 1 , . . . , x n , y 1 , . . . , y n are real n um b ers, as ω is a quadrati in teger, note that n Y i =1 ( x i + ω y i ) = P ( x 1 , . . . , x n , y 1 , . . . , y n ) + ω Q ( x 1 , . . . , x n , y 1 , . . . , y n ) where P and Q are p olynomials with in teger o eien ts in x 1 , . . . , x n , y 1 , . . . , y n . By writing ea h o eien t of a n × n omplex matrix X in the basis (1 , ω ) o v er R , the equation det X = 1 giv es a system of t w o p olynomial equations with in teger o eien t, with unkno wn the o ordinates of the o eien ts of X in (1 , ω ) . Hene, there exists an algebrai group G dened o v er Q su h that G ( Z ) = SL 2 () , G ( Q ) = SL 2 ( K − m ) and G ( R ) = SL 2 ( C ) . As the Lie group G ( R ) is onneted and semisim- ple, with asso iated symmetri spae the real h yp erb oli 3 -spae, the algebrai group G is onneted, semisimple with R -rank one. Let P b e the algebrai subgroup of G orresp ond- ing to the upp er triangular subgroup of 2 × 2 matries, so that P is the stabilizer of the p oin t at innit y ∞ in the upp er halfspae mo del of H 3 R . Let Γ R , I b e the nite index subgroup of the group SL 2 () , whi h is the preimage in the group morphism SL 2 () → SL 2 ( / I ) of redution mo dulo I of the subgroup of upp er triangular matries. By Theorem 6.9, the subgroup Γ R , I is a lattie in SL 2 ( C ) , and its set of parab oli xed p oin ts is P Γ R , I = SL 2 K − m · ∞ = K − m ∪ {∞} , 79 as  0 − 1 1 0  and  x 0 1 x − 1  , for ev ery x in K − m − { 0 } , are elemen ts of SL 2 ( K − m ) sending ∞ to 0 and x resp etiv ely . Note that if I = = − m , then Γ R , I = PSL 2 ( − m ) is a Bian hi group, whi h is w ell- kno wn to b e a lattie in PSL 2 ( C ) (see for instane [MR ℄). The fat that P Γ R , − m = K − m ∪ {∞} is also pro v en in [ EGM , Prop. 2.2, page 314℄. (2) Reall that N ( ω ) = ω ω and T r( ω ) = ω + ω = 2 Re ω are in tegers, as ω is an algebrai in teger. If x, y , x ′ , y ′ are real n um b ers, note that ( x + ωy )( x ′ + ω y ) = ( xx ′ + N ( ω ) y y ′ + T r( ω ) y x ′ ) + ω ( xy ′ − yx ′ ) . Reall that the matrix Q (in tro dued in Setion 6.1 ) has in teger o eien ts. Hene b y writing ea h o eien t of a ( n + 1) × ( n + 1) omplex matrix X in the basis (1 , ω ) o v er R , the system of equations giv en b y det X = 1 and X ∗ QX = Q b eomes a system of 2(( n + 1) 2 + 1) p olynomial equations with in teger o eien ts, with unkno wn the o ordinates of the o eien ts of X in (1 , ω ) . Therefore there exists an algebrai group G dened o v er Q su h that G ( Z ) = S U Q () , G ( Q ) = SU Q ( K − m ) and G ( R ) = SU Q . As the Lie group G ( R ) is onneted and semisim- ple, with asso iated symmetri spae the omplex h yp erb oli n -spae, the algebrai group G is onneted, semisimple with R -rank one. Let P b e the algebrai subgroup of G or- resp onding to the upp er triangular subgroup of ( n + 1) × ( n + 1) matries, so that P is the stabilizer of the p oin t at innit y ∞ in the Siegel domain mo del of H n C , or of the p oin t [1 : 0 : 0] in the pro jetiv e mo del. By Theorem 6.9 , the group Γ C , I dened in Setion 6.1 is a lattie in SU Q , and its set of parab oli xed p oin ts is P Γ C , I = SU Q ( K − m ) · ∞ . By Witt's theorem, SU Q ( K − m ) ats transitiv ely on the isotropi lines in K − m n − 1 for the hermitian form q . Hene P Γ C , I is exatly the set of rational p oin ts of the quadri o v er K − m with equation q = 0 in P n ( C ) . If n = 2 and = − m , w e reo v er Prop osition 6.5 (1). (3) Let I ′ b e a non-zero t w o-sided ideal in an order ′ of a quaternion algebra A ( Q ) o v er Q su h that A ( Q ) ⊗ Q R = H . F or ev ery eld K on taining Q , dene A ( K ) = A ( Q ) ⊗ Q K . Let ( e 1 , e 2 , e 3 , e 4 ) b e a basis of ′ as a Z -mo dule. It is also a basis of A ( K ) as a K -v etor spae for ev ery eld K on taining Q . If x = x 1 + x 2 i + x 3 j + x 4 k is an elemen t in H , written in the standard basis (1 , i, j, k ) , let T r x = 2 x 1 b e its redued trae, and N ( x ) = x 2 1 + x 2 2 + x 2 3 + x 2 4 b e its redued norm. A 2 × 2 matrix X with o eien ts in H has Dieudonné determinan t 1 if and only if N ( ad ) + N ( bc ) + T r( acdb ) = 1 (- 35 -) (see for instane [Kel , page 1084℄). The maps R 4 → R dened b y ( x 1 , x 2 , x 3 , x 4 ) 7→ N ( x 1 e 1 + x 2 e 2 + x 3 e 3 + x 4 e 4 ) and ( x 1 , x 2 , x 3 , x 4 ) 7→ T r( x 1 e 1 + x 2 e 2 + x 3 e 3 + x 4 e 4 ) are p olynomial maps in x 1 , x 2 , x 3 , x 4 with rational o eien ts. By writing ea h o eien t a, b, c, d of a 2 × 2 matrix X with o eien ts in A ( K ) in the basis ( e 1 , e 2 , e 3 , e 4 ) for an y eld K , the equation (- 35 - ) b eomes a p olynomial equation with o eien ts in Q , with unkno wn the o ordinates of the o eien ts of X in ( e 1 , e 2 , e 3 , e 4 ) . Hene there exists an algebrai group G dened o v er Q su h that G ( Z ) = SL 2 ( ′ ) and G ( K ) = SL 2 ( A ( K )) for ev ery eld K on taining Q . As the Lie group G ( R ) = S L 2 ( H ) is onneted and semisimple, with asso iated symmetri spae the real h yp erb oli 5 -spae, 80 the algebrai group G is onneted, semisimple with R -rank one. Let P b e the algebrai subgroup of G orresp onding to the upp er triangular matries, so that P is the stabilizer of the p oin t at innit y ∞ in the upp er halfspae mo del of H 5 R . Let Γ I ′ b e the group in tro dued in Setion 6.2 , whi h has nite index in SL 2 ( ′ ) . By Theorem 6.9, the subgroup Γ I ′ is a lattie in SL 2 ( H ) , and its set of parab oli xed p oin ts is P Γ I ′ = SL 2 ( A ( Q )) · ∞ . As A ( Q ) is a division algebra, the same argumen t as for example (1) sho ws that P Γ I ′ = A ( Q ) ∪ {∞} . 6.4 The ubiquit y of Hall ra ys In this subsetion, w e giv e appliations of our geometri results from Setion 5 to the framew ork of Diophan tine appro ximation in negativ ely urv ed manifolds, in tro dued in [HP3 , HP4℄, to whi h w e refer for notation and ba kground. In partiular, w e will onsider arithmetially dened examples. See also the previous w orks of [F or , Ser℄, among man y others. Let M b e a omplete, nonelemen tary , geometrially nite Riemannian manifold with setional urv ature at most − 1 and dimension at least 3 . Let π : f M → M b e a univ ersal Riemannian o v ering, with o v ering group Γ . Let e b e a usp of M , and, as in Setion 5.1 , let V e b e a xed Margulis neigh b orho o d of e , H e a horoball in f M with π ( H e ) = V e and ξ e the p oin t at innit y of H e . Note that requiring V e to b e a Margulis neigh b orho o d is equiv alen t to requiring H e to b e preisely in v arian t under Γ . In the previous w orks [HP3 , HP4℄, it w as required that V e is the maximal Margulis neigh b orho o d, as this mak es the onstrutions indep enden t of the  hoie of V e . But as it is not alw a ys easy to determine the maximal Margulis neigh b orho o d of a usp, and as it is not neessary for the statemen ts, w e will x some  hoie of V e (or equiv alen tly H e ) whi h is not neessarily maximal. Three (lasses of ) examples. These examples ould in fat b e orbifolds rather than manifolds, but the extension to this on text is ob vious. W e use the same notation as in the examples of Subsetion 6.3. (1) Let Γ R , I b e the nite index subgroup of the group SL 2 () , whi h is the preimage, b y the group morphism SL 2 () → SL 2 ( / I ) of redution mo dulo I , of the subgroup of upp er triangular matries. The quotien t M = Γ R , I \ H 3 R is a nite v olume real h yp erb oli orbifold. Let π : H 3 R → M b e the anonial pro jetion, e the usp of M orresp onding to ξ e = ∞ , and H e b e the horoball of p oin ts of Eulidean heigh t at least 1 . As  1 1 0 1  b elongs to Γ R , I , it is w ell kno wn that H e is preisely in v arian t under Γ R , I . F urthermore, if I = , then  0 1 − 1 0  b elongs to Γ R , I , hene H e is maximal. F or more details, see [HP3 ℄, end of Setion 5. (2) Let Γ C , I b e the group of isometries of the Siegel domain mo del H n C of the omplex h yp erb oli n -spae with (onstan t) holomorphi setional urv ature − 1 , that w as in tro- dued in Setion 6.1 . Let M b e the nite v olume omplex h yp erb oli orbifold Γ C , I \ H n C , whi h is endo w ed with the quotien t of the renormalized Riemannian distane d ′ H n C in order for its setional urv atures to b e at most − 1 . Let π : H n C → M b e the anonial pro jetion, e b e the usp of M orresp onding to ξ e = ∞ , and H e the horoball H 2 Im ω if Re ω ∈ Z , and H 4 Im ω otherwise, whi h is preisely in v arian t under Γ C , I b y Lemma 6.4 (and maximal if I = = − 1 ). 81 (3) Let I ′ b e a non-zero t w o-sided ideal in an order ′ of a quaternion algebra A ( Q ) o v er Q su h that A ( Q ) ⊗ Q R = H , and Γ I ′ the group of isometries of the upp er halfspae mo del H 5 R of the real h yp erb oli 5 -spae with (onstan t) setional urv ature − 1 , that w as in tro dued in Setion 6.2 . Let M b e the nite v olume real h yp erb oli orbifold Γ I ′ \ H 5 R , π : H 5 R → M b e the anonial pro jetion, e b e the usp of M orresp onding to ξ e = ∞ , and H e the horoball H 1 , whi h is preisely in v arian t under Γ I ′ b y Lemma 6.8 (and maximal if I ′ = ′ ). Let the link of e in M , Lk e = Lk e ( M ) , b e the spae of lo ally geo desi lines (up to translation at the soure) starting from e in M that are non w andering (i.e. su h that ea h of them aum ulates in some ompat subset of M ). Let Rat e b e the spae of lo ally geo desi lines starting from e and on v erging to e . Normalize the lo ally geo desi lines in Lk e ∪ R at e so that their rst in tersetion with ∂ V e is at time 0 . Endo w Lk e ∪ R at e with the ompat- op en top ology . Let ΛΓ ⊂ ∂ ∞ f M b e the limit set of Γ , P Γ ⊂ ΛΓ b e the set of parab oli xed p oin ts of Γ , and Γ ∞ b e the stabilizer of ξ e in Γ . Then the maps Γ ξ e − { ξ e } → Rat e and ΛΓ − P Γ → Lk e , whi h asso iate to x the pro jetion in M b y π of the geo desi line starting from ξ e and ending at x , indue a bijetion Γ ∞ \ (Γ ξ e − { ξ e } ) → Rat e and a homeomorphism Γ ∞ \ (ΛΓ − P Γ ) → Lk e . W e iden tify these spaes b y these maps. Note that Lk e ∪ Rat e is ompat if and only if M has only one usp, and that Rat e is dense in Lk e ∪ Rat e (as Γ ξ e is dense in ΛΓ ). Diophan tine appro ximation in M (see [HP3, HP4 , HP5℄) studies the rate of on v ergene of sequenes of p oin ts in Rat e to giv en p oin ts in Lk e . F or ev ery r in Rat e , let D ( r ) , alled the depth of r , b e the length of the subsegmen t of r b et w een the rst and the last meeting p oin ts with ∂ V e . Examples. (1) Consider M = Γ R , I \ H 3 R . Then P Γ R , I ⊂ C ∪ {∞} is exatly K − m ∪ {∞} , b y the example (1) of Setion 6.3 . Th us, Lk e ( M ) = (Γ R , I ) ∞ \ ( C − K − m ) . In a omm utativ e unitary ring R , w e denote b y h p 1 , . . . , p k i the ideal generated b y p 1 , . . . , p k ∈ R . It is easy to pro v e (see for instane [EGM, Lem. 2.1, page 314℄) that Rat e is the set of elemen ts r = p/q (mo dulo (Γ R , I ) ∞ ) with ( p, q ) ∈ × I su h that h p, q i = . F urthermore (see [HP3, Lem. 2.10℄) D ( r ) = 2 log | q | . (2) Consider M = Γ C , I \ H n C . Let Q ( R ) b e the real quadri ∂ ∞ H n C − {∞} . By onsider- ing a basis of K − m o v er Q , it is easy to see that Q ( R ) is the set of R -p oin ts of a quadri Q dened o v er Q (whi h dep ends on m ), whose set Q ( Q ) of Q -p oin ts is Q ( R ) ∩ ( K − m × K n − 1 − m ) . W e ha v e Lk e = (Γ C , I ) ∞ \ ( Q ( R ) − P Γ C , I ) . By the example (2) of Setion 6.3 , w e ha v e P Γ C , I = Q ( Q ) ∪ {∞} . Then Rat e is the quotien t mo dulo (Γ C , I ) ∞ of the subset of Q ( Q ) of p oin ts of the form ( a/c, α/c ) with ( a, α, c ) ∈ × I n − 1 × I su h that there exist b, c, β , γ , δ, A matries of the appropriate size su h that   a γ ∗ b α A β c δ ∗ d   b elongs to Γ C , I . By Prop osition 6.5 (2), this existene requiremen t is equiv alen t to the requiremen t that q ( a, α, c ) = 0 and h a, α, c i = , if n = 2 and = − m . By Prop osition 6.5 (3), Rat e = (Γ C , I ) ∞ \ Q ( Q ) if n = 2 , I = = − m and m = 1 , 2 , 3 , 7 , 11 , 19 , 43 , 67 , 163 . 82 If r ∈ Rat e is of the form ( a/c, α/c ) (mo dulo (Γ C , I ) ∞ ) as ab o v e, then b y Lemma 6.3, w e ha v e D ( r ) = log | c | +  log Im ω if Re ω ∈ Z , log(2 Im ω ) otherwise . (3) Consider M = Γ I ′ \ H 5 R . W e ha v e P Γ I ′ = A ( Q ) ∪ {∞} , b y the example (3) of Setion 6.3. Hene Lk e = (Γ I ′ ) ∞ \ ( H − A ( Q )) . It is easy to see that Rat e is the set of elemen ts r = p q − 1 (mo dulo (Γ I ′ ) ∞ ) with ( p, q ) ∈ ′ × ( I ′ − { 0 } ) su h that there exists r , s ∈ ′ with | q r − q pq − 1 s | = 1 . F urthermore, b y Lemma 6.7 , w e ha v e D ( r ) = 2 log | q | . The uspidal distan e d ′ e ( γ , γ ′ ) of γ , γ ′ in Lk e ∪ Rat e is the minim um of the e d ′ e ( e γ , e γ ′ ) for e γ , e γ ′ t w o lifts of γ , γ ′ to f M starting from ξ e , where e d ′ e ( e γ , e γ ′ ) is the greatest lo w er b ound of r > 0 su h that the horosphere en tered at e γ (+ ∞ ) , at signed distane − log 2 r from ∂ H e on the geo desi line ] ξ e , e γ (+ ∞ )[ , meets e γ ′ (see [HP3, Set. 2.1℄). Though not neessarily an atual distane, e d ′ e is equiv alen t to the Hamenstädt distane (see Subsetion 3.1 and [HP3 , Rem. 2.6℄). Examples. (1) If M has onstan t urv ature − 1 , if one iden ties Lk e ∪ Rat e with a subset of ∂ H e b y the rst in tersetion p oin t, then d ′ e is the indued Riemannian distane on ∂ H e , whi h is Eulidean (see [HP3, Set. 2.1℄); in partiular, if M = Γ R , I \ H 3 R or M = Γ I ′ \ H 5 R , then d ′ e is the quotien t of the standard Eulidean distane on Lk e ∪ Rat e iden tied with a subset of (Γ R , I ) ∞ \ C or (Γ I ′ ) ∞ \ H . (2) If M is Hermitian with onstan t holomorphi setional urv ature − 1 , then d ′ e is no longer Riemannian, but b y Prop osition 6.2 is a m ultiple of the mo died Cygan distane d ′ Cyg . In partiular, if M = Γ C , I \ H n C , then d ′ e is the quotien t b y (Γ C , I ) ∞ ating on ∂ ∞ H n C of the distanes ( 1 2 √ Im ω d ′ Cyg if Re ω ∈ Z 1 2 √ 2 Im ω d ′ Cyg otherwise . Remark. The laim in the rst paragraph of Setion 3.11 in [ HP4 ℄ (where w e only on- sidered m = 1 and I = = − 1 ) that the uspidal distane oinides with the Hamenstädt distane is inorret; as the Cygan distane and the mo died Cygan distane are equiv a- len t, this do es not  hange the statemen t of the main results, Theorems 3.1, 3.2, 3.4, and 3.5 of [HP4 ℄. As d ′ Cyg ≤ √ 2 d Cyg so that d Cyg ≥ √ 2 d ′ e , the onstan t 1 4 √ 5 app earing in Theorem 3.6 of [HP4 ℄ has to b e replaed b y 1 2 4 √ 5 . With M as in the b eginning, for ev ery x in Lk e , dene the appr oximation  onstant c ( x ) of x as c ( x ) = lim inf r ∈ Rat e , D ( r ) →∞ d ′ e ( x, r ) e D ( r ) . The L agr ange sp e trum of M with resp et to e is the subset Sp Lag ( M , e ) of R onsisting of the onstan ts c ( x ) for x in Lk e . It is sho wn in [HP3 ℄ that • c ( x ) is w ell dened (as Rat e is dense in Lk e ∪ Rat e and { D ( r ) : r ∈ Rat e } is a disrete subset of R with nite m ultipliities), 83 • c ( x ) is nite (as x is non w andering), (Note that if y is a lo ally geo desi line starting from e in M that on v erges in to a usp of M , then the same form ula w ould yield c ( y ) = + ∞ .) • the least upp er b ound of Sp Lag ( M , e ) , denoted b y K M ,e and alled the Hurwitz  on- stant , is nite. In partiular, Sp Lag ( M , e ) ⊂ [0 , K M ,e ] . The follo wing result tells us that the Lagrange sp etrum on tains a non trivial initial in terv al [0 , c ] , with a univ ersal lo w er b ound on c (whose optimal v alue w e do not kno w). Theorem 6.10 L et M b e a  omplete, nonelementary, ge ometri al ly nite R iemannian manifold with se tional urvatur e at most − 1 and dimension at le ast 3 , and let e b e a usp of M . The L agr ange sp e trum Sp Lag ( M , e )  ontains the interval [0 , 0 . 033 7] . In p artiular, K M ,e ≥ 0 . 0337 . Pro of. By [HP3 ℄, the map x 7→ − 2 log x sends bijetiv ely the Lagrange sp etrum on to the asymptoti heigh t sp etrum. W e apply Corollary 5.15 and the omputation ab o v e it.  A preise v ersion of this theorem is stated as orollaire 5 in [ PP2 ℄ when M is a real or omplex h yp erb oli manifold. Theorem 1.7 in the in tro dution follo ws immediately , b y the rst example disussed in this setion. By v arying the (non uniform) arithmeti latties in the isometry group of a negativ ely urv ed symmetri spae (see for instane [MR, MWW ℄), other arithmeti appliations are p ossible. W e only state t w o of them in what follo ws. Let I ′ b e a non-zero t w o-sided ideal in an order ′ of a quaternion algebra A ( Q ) o v er Q ramifying o v er R , and N the redued norm on A ( R ) = A ( Q ) ⊗ Q R (see for instane [Vig℄, and Setion 6.2 ). F or ev ery x ∈ A ( R ) − A ( Q ) , dene the appr oximation  onstant of x b y c ( x ) = lim inf ( p,q ) ∈ ′ × I ′ : ∃ r,s ∈ ′ N ( q r − q pq − 1 s )=1 , N ( q ) →∞ N ( q ) N ( x − pq − 1 ) 1 2 , and the Hamilton-L agr ange sp e trum for the appro ximation of elemen ts of H b y elemen ts of ′ I ′− 1 as the subset of R onsisting of the c ( x ) for x ∈ A ( R ) − A ( Q ) . Note that c ( x ) is nite if x / ∈ A ( Q ) , as then x is not a parab oli xed p oin t of Γ I ′ . Apply Theorem 6.10 to M = Γ I ′ \ H 5 R with the ab o v e disussions of the third example to get the follo wing result. Theorem 6.11 The Hamilton-L agr ange sp e tr a  ontain the interval [0 , 0 . 033 7] .  In the ase when I ′ = ′ and ′ is the Hurwitz maximal order in Hamilton's quaternion algebra A ( Q ) ⊂ H (see Subsetion 6.2), A. S hmidt [S h2 ℄ pro v ed that the Hamilton- Lagrange sp etrum on tains √ 2 S p Q where Sp Q is the lassial Lagrange sp etrum for the appro ximation of real n um b ers b y rational n um b ers. As Sp Q on tains [0 , µ ] where µ is F reiman's onstan t (see the end of Subsetion 5.4 ), this pro v es that the Hamilton-Lagrange sp etrun in this ase on tains the in terv al [0 , 0 . 312 ] , whi h is reasonably lose to our general estimate. Note that the fat that our appro ximation onstan t oinides with the in v erse of A. S hmidt's appro ximation onstan t follo ws from [S h1 , Thm.5℄ Let m b e a squarefree p ositiv e in teger, I b e a non-zero ideal in an order in the ring of in tegers − m of the imaginary quadrati n um b er eld Q ( i √ m ) , and ω b e the elemen t of − m with Im ω > 0 su h that = Z + ω Z . Let E , I b e the set of ( a, α, c ) in × I n − 1 × I 84 su h that there exists a matrix of the form   a γ ∗ b α A β c δ ∗ d   that b elongs to Γ C , I . If n = 2 and = − m , then, as seen previously , E , I = { ( a, α, c ) ∈ × I n − 1 × I : q ( a, α, c ) = 0 , h a, α, c i = } . W e do not kno w if this is the ase for ev ery n, I , . F or ev ery x ∈ Q ( R ) − Q ( Q ) , dene the appr oximation  onstant of x b y c ( x ) = lim inf ( a,α,c ) ∈ E , I , | c |→∞ | q | d ′ Cyg ( x, ( a/c, α/c )) , and the Heisenb er g-L agr ange sp e trum , for the appro ximation of elemen ts of Q ( R ) b y elemen ts of { ( a/c, α/c ) : ( a, α, c ) ∈ E , I } ⊂ Q ( Q ) , as the subset of R onsisting of the c ( x ) for x ∈ Q ( R ) − Q ( Q ) . Note that c ( x ) is nite if x / ∈ Q ( Q ) , as then x is not a parab oli xed p oin t of Γ C , I . Our last result follo ws from Theorem 6.10 and the previous disussions of the seond example. 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Bo x 35 40014 Univ ersit y of Jyv äskylä, FINLAND. e-mail: p arkkonemaths.jyu. Départemen t de Mathématique et Appliations, UMR 8553 CNRS Eole Normale Sup érieure, 45 rue d'Ulm 75230 P ARIS Cedex 05, FRANCE e-mail: F r e deri.Paulinens.fr 89