A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry

A c haracterization of i rreducible symmetric spaces and Euclidean buildings of hi gher rank b y their asymptotic geometry Bernhard Leeb June 17, 199 7 Abstract. W e study geo desically complete and lo cally compact Hadamard spaces X whose Tits boundary is a connected irreducible spherical building. W e sho w that X is symmetric iﬀ complete geo desics in X do not branch and a Euclidean building otherwise. F urthermore, ev ery b oundary eq uiv alence (cone top ology homeomorphism preserving the Tits metric) b et w een t w o suc h spaces is induced b y a homothety . As an application, w e can extend the M ostow and Prasad rigidit y theorems to com- pact singular (orbi)spaces of nonp ositive curv ature which a re homo t o p y equiv alen t to a quotient of a symmetric space or Euclidean building by a co compact group of isometries. Con tents 1 In tro duction 2 1.1 Main result, bac kground, motiv ation and an application . . . . . . . . 2 1.2 Around the argumen t . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Preliminaries 8 2.1 Hadamard spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Filling spheres at inﬁnit y by ﬂats . . . . . . . . . . . . . . . . 8 2.1.2 Con v ex cores . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 Spaces of strong asymptote classes . . . . . . . . . . . . . . . 11 2.1.4 T yp es of isometries . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Visibilit y Hadamard spaces . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Buildings: Deﬁnition, v o cabulary and examples . . . . . . . . . . . . 13 2.3.1 Spherical buildings . . . . . . . . . . . . . . . . . . . . . . . . 1 3 2.3.2 Euclidean buildings . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Lo cally compact top ological groups . . . . . . . . . . . . . . . . . . . 15 3 Holonom y 16 4 Rank one: Rigidity of highly symmetric visibilit y spaces 20 4.1 General prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Butterﬂy construction of small axial isometries . . . . . . . . . . . . . 21 1 4.3 The discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3.1 Equiv a rian t rigidit y for trees . . . . . . . . . . . . . . . . . . . 24 4.4 The non-discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5 Geo desically complete Hadamard spaces w it h building b oundary 26 5.1 Basic prop erties of parallel sets . . . . . . . . . . . . . . . . . . . . . 26 5.2 Boundary isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.3 The case of no branc hing . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.4 The case of branc hing . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.4.1 Disconnectivit y o f F ¨ ursten b erg b o undary . . . . . . . . . . . . 34 5.4.2 The structure of parallel sets . . . . . . . . . . . . . . . . . . . 36 5.4.3 Pro of of the main result 1.2 . . . . . . . . . . . . . . . . . . . 36 5.5 Inducing b oundary isomorphisms by homotheties : Proof of 1.3 . . . . 38 5.6 Extension of Mosto w and Prasad Rigidity to singular spaces of non- p ositiv e curv ature: Pro of of 1.5 . . . . . . . . . . . . . . . . . . . . . 40 Bibliograph y 41 1 In tro duction 1.1 Main result, b ackgro un d, motiv ation and an app lication Hadamard manifolds are simply-connected complete Riemannian manifolds of non- p ositiv e sectional curv ature. Prominen t examples are Riemannian symmetric spaces of noncompact ty p e, but many m ore examples o ccur as univ ersal co v ers of closed non- p ositiv ely curv ed manifolds. F or instance, most Haken 3-manif olds admit metrics of nonp ositiv e curv ature [Le95]. Not the notion of sectional curv ature itself, how ev er the notion of a n upp er curv ature b ound can b e expressed purely b y inequalities inv olv- ing the distances b et w een ﬁnitely man y p oints but no deriv ativ es of the Riemannian metric, a nd hence generalizes from the narrow w orld of Riemannian manifolds to a wide class of metric spaces, cf. [Al57]. The natural g eneralization of Hadamard manifolds are Hadamard spaces, i.e. complete geo desic metric spaces whic h are non- p ositiv ely curv ed in the (globa l) sense of distance comparison, see [Ba95, KL 9 6]. Hadamard spaces comprise b esides Ha da mard manifolds a large class of in teresting singular spaces , among them Euclidean buildings (the discrete cousins of symmetric spaces), man y piecewise Euclidean or Riemannian complexes o ccuring, for instance, in geometric group theory , and branc hed co vers of Hadamard manifolds. Hadamard spaces receiv ed m uc h atten tion in the last decade, notably with view t o geometric group theory , a main imp etus coming f rom Gromo v’s w ork [Gr 8 7, Gr93]. W e recall that a f undamen tal feature of a Hadamard space is the con v exit y of its distance function with the drastic consequences suc h a s uniqueness o f geo desics and in particular con tra ctibilit y . This illustates that already geodesics, undoubtedly fundamen tal o b jects in geometric considerations, are rather w ell-b eha v ed a nd their b eha vior can b e to some exten t con trolled, whic h gets the f o ot in the do or for a mor e adv anced geometric understanding. The imp ortance of the geometry of nonp ositiv e curv ature lies in the coincidence that one has a ric h supply of in teresting examples reac hing in to many diﬀeren t branc hes of mathematics (lik e geometric group theory , 2 represen tation theory , arithmetic) a nd, at the same time, these spaces share simple basic geometric properties whic h mak es them understandable to a certain exten t and in a uniform w ay . W e will be in terested in asymptotic informatio n and the restrictions which it im- p oses o n the geometry of a Hadamard space X . This is related to the rigidit y ques- tion, already classical in global Riemannian geometry , ho w top ological prop erties of a (for instance closed) Riemannian manifold with certain lo cal (curv ature) constraints are reﬂected in its geometry 1 and b elow (1.3) w e will presen t an application in this direction. Let us ﬁrst describ e whic h asymptotic infor mation w e consider. The g e ometric or ide al b oundary ∂ ∞ X of a Hadama r d space X is deﬁned as t he set o f equiv alence classes of asymptotic geo desic rays. 2 The top ology on X extends to a natur a l c one top olo gy on the geometric completion ¯ X = X ∪ ∂ ∞ X whic h is compact iﬀ X is lo cally compact. The ideal b oundar y p oin ts ξ ∈ ∂ ∞ X can b e though t of as the w ays to go straigh t to inﬁnit y 3 . It is fair to say that the top olog ical t yp e o f ∂ ∞ X is not a v ery strong in v arian t, for example it is a ( n − 1)-sphere for any n -dimensional Hadamard manifold. Besides the cone t o p ology there is a nother in teresting structure on ∂ ∞ X , namely the Ti ts an g l e metric in tro duced b y Gromo v in full generalit y in [BGS8 5]. F or t wo p oin ts ξ 1 , ξ 2 ∈ ∂ ∞ X at inﬁnit y their Tits angle ∠ T its ( ξ 1 , ξ 2 ) measures the maximal visual angle ∠ x ( ξ 1 , ξ 2 ) under whic h t hey can b e seen from a p oint x inside X , or equiv a len tly , it measures the asymptotic linear rate at whic h unit sp eed g eo desic ra ys ρ i asymptotic to the ideal po in ts ξ i div erge from eac h other. If X has a strictly negativ e curv ature b ound the Tits b oundary ∂ T its X = ( ∂ ∞ X , ∠ T its ) is a discrete metric space and only of mo dest in terest. How ev er, if X features substructures of extremal curv ature zero, suc h a s ﬂats, i.e. con v ex subsets isometric to Euclidean space, then connected comp onen ts appear in t he Tits bo undary and the Tits metric b ecomes an in teresting structure. 4 The cone top olog y to gether with the Tits metric on ∂ ∞ X a re the asymptotic data whic h w e consider here. Our results ﬁnd shelte r under the ro of of the follo wing: Meta-Question 1.1 What ar e the implic ations of these asymp totic d ata for the ge- ometry of a Hadamar d sp ac e? The main result is t he following c haracterization of symmetric spaces and Eu- clidean buildings of higher rank as Hadamard spaces with spherical building b ound- ary: 1 Since the universal cover is contractible, the entire top o logical information is contained in the fundamen tal gro up and one ca n ask which of its alge braic prop erties are visible in the g eometry . 2 F or tw o unit speed geo des ic rays ρ 1 , ρ 2 : [0 , ∞ ) the distance d ( ρ 1 ( t ) , ρ 2 ( t )) of trav ellers along the rays is a convex function. If it is bo unded (and hence non-incr easing) the r ays are called asymptotic . 3 Examples: The geometr ic completion o f hyperb olic plane can b e obtained by taking the closure in the Poincar ´ e disk mo de l; one obtains the Poincar ´ e Compact Disk mo de l. T he geo metric boundar y of a metric tree is the set o f its ends which is a Cantor set if it has no isolated points. 4 ∂ T its S L (3 , R ) /S O (3) is the 1-dimensional spherical building asso ciated to the real pro jective plane. 3 Main Theorem 1.2 L et X b e a lo c al ly c omp act Hadamar d sp ac e with extendible ge o desic se gments 5 and assume that ∂ T its X is a c onn e cte d thick irr e ducible spheric al building. Then X is a Riemannian symm etric sp ac e or a Euclide an building. In the smo oth case, i.e. for Hadamard manifolds, 1.2 follo ws f rom w or k of Ball- mann and Eb erlein, cf. [Eb88, Theorem B], o r else from argumen ts of Gromov [BGS85] and Burns-Spatzier [BS87]. There is a dic hoto m y in to tw o cases, according to whether geo desics in X branc h o r not. In the absence of branc hing the ideal b oundaries are v ery symmetric b ecause there is an in volution ι x of ∂ ∞ X at ev ery p oint x ∈ X , and one can a dapt argumen ts fro m G romo v in the pro of o f his rigidit y theorem [BGS85]. Our main contribution lies in the case of geodesic branc hing. There the boundary at inﬁnit y admits in general no non-trivial sym metries a nd a no ther a pproac h is needed. W e sho w moreo v er that fo r the spaces considered in 1.2 t he extreme situation o ccurs that X is completely determined b y its asymptotic data up to a scale factor : Addendum 1.3 L et X b e a symmetric sp a c e or a thick Euclide an building, irr e- ducible and of r ank ≥ 2 , and let X ′ b e another such sp ac e. The n any b oundary isomorphism ( c one top olo gy home omorph i s m pr eserving the Tits metric) φ : ∂ ∞ X → ∂ ∞ X ′ (1) is induc e d by a hom othety. 1.3 fo llo ws from Tits classiﬁcation for automorphisms of spherical buildings in the cases when X has man y symmetries, e.g. when it is a Riemannian symmetric space or a Euclidean building a sso ciated to a simple algebraic group o v er a lo cal ﬁeld with non-arc himedean v aluatio n. This is in particular true if r ank ( X ) ≥ 3 ho we ver it do es not cov er the cases when X is a rank 2 Euclidean building with small isometry group. Our metho ds provide a uniform pro of in all cases and in particular a direct argumen t in the symmetric cases. A main motiv ation for us was Mosto w’s Strong Rigidit y Theorem for lo cally sym- metric spaces, namely the irreducible case of higher rank: Theorem 1.4 ([Mos73]) L et M and M ′ b e lo c al ly symmetric sp ac es whose universal c overs ar e irr e ducib l e symmetric sp ac es of r ank ≥ 2 . Then any isomorphism π 1 ( M ) → π 1 ( M ′ ) of fundamental gr oups is ind uc e d by a homothety M → M ′ . It is natural to ask whether lo cally symme tric spaces are rigid in the wider class of closed manifolds of nonp ositiv e sectional curv at ure. This is true and the conte nt of Gro mo v’s Rigidity Theorem [BGS85]. As an application of our main results w e presen t an extens ion of Mosto w’s theorem a s w ell as Prasad’s analogue f or compact quotien ts of Euclidean buildings [Pra79] to the larger class of singular nonp ositiv ely curv ed (orbi)spaces: Application 1.5 L et X b e a lo c al ly c omp act Hadama r d sp ac e with extendible ge o de sic se gmen ts and let X model b e a symmetric sp ac e (of nonc omp act typ e) or a thick Eu- clide an building. Supp ose f urthermor e that al l irr e ducible factors of X model have r a nk 5 I.e. ev er y geo desic segment is con tained in a complete geo des ic. 4 ≥ 2 . I f the sam e ﬁnitely gener ate d gr oup Γ acts c o c omp actly and pr op erly disc ontin- uously on X and X model then, after suitably r es c aling the metrics on the irr e ducible factors of X model , ther e is a Γ -e quivariant isometry X → X model . I.e. among (p ossibly singular) geo desic ally complete compact spaces of nonp ositiv e curv ature (in the lo cal sense), quotien ts of irreducible higher rank symmetric spaces or Eulidean buildings are determined by their ho mo t o p y type. Example 1.6 On a lo c al ly symmetric sp a c e with irr e ducible hig h er r ank universal c over ther e exists no pie c ew i s e Euclide an singular m etric of nonp ositive curvatur e. As w e said, 1.5 is due to G romo v [BGS85] in the case that X is smo oth Rieman- nian. Although w e extend Gromo v’s extens ion o f Mosto w Rigidity further to singular spaces, the news of 1.5 lie mainly in the building case. Ac knowled gemen t s. I w ould lik e to use this opp o rtunit y to thank the Son- derforsc h ungsb ereic h SFB 256 and W erner Ballmann for generously supp o r t ing m y mathematical researc h o v er the past y ears. I mak e extens ive use of the geometric approac h to the theory of generalized Tits buildings dev elop ed in [KL9 6] together with Bruce Kleiner and thank him for numer- ous discussions related to the sub ject of this pap er. I a m grateful to W erner Ballmann and Misha Kap ovic h for po in ting out coun tably many errors in a la te version of the man uscript, to Jens Heb er fo r comm unicating a reference and to the Complutense library in Madrid for sev eral crucial suggestions. Part of this w ork w as done while visiting the Instituto de Matematicas de la UNAM, Unidad Cuerna v aca, in Mex- ico. I thank the institute for its great ho spitality , especially Pepe Seade and Alb erto V erjo vski. 1.2 Around the argumen t In this section I attempt to describ e the scenery around the pro ofs of 1.2 and 1 .3 for the ra nk 2 case, i.e. when maximal ﬂat s in X hav e dimension 2 and ∂ T its X is 1-dimensional. The rank 2 case is a n yw ay the critical case because there the rigidit y is qualitativ ely w eake r than in the case of rank ≥ 3. This diﬀerence is reﬂecte d in Tits classiﬁcation theorem f or spherical buildings [Ti74] whic h asserts, roughly sp eaking, that all thic k irreducible spherical buildings of rank ≥ 3 (that is, dimension ≥ 2) are canonically attached to simple alg ebraic or classical groups. In contrast there exist uncoun tably man y absolutely asymme tric 1-dimensional spherical buildings, for example those corresponding to exotic pro jectiv e planes, and uncoun tably man y of them o ccur as Tits b oundaries of rank 2 Euclidean buildings with trivial isometry group. F or a singular geo desic l in X , i.e. a geo desic asymptotic to v ertices in ∂ T its X , w e consider the union P ( l ) of a ll geo desics parallel to l and its cross section C S ( l ) whic h is a lo cally compact Hada mard space with discrete Tits b oundary (a s for a rank 1 space). 6 F or t w o asymptotic geo desics l and l ′ one can canonically iden tify the ideal 6 F or instance, if X = S L (3 , R ) /S O (3) then the cross sections of singular geo des ic s ar e hype rb olic planes. More generally , if X is a symmetric space of rank 2 then these cross se c tions a re rank -1 symmetric s pa ces. If X is a Euclidean building of r ank 2 they are rank-1 Euclidea n buildings, i.e. metric trees. 5 b oundaries ∂ ∞ C S ( l ) and ∂ ∞ C S ( l ′ ). A prio ri this iden tiﬁcation o f b oundaries do es not extend to an isometry b etw een the cross sections, but it extends to an isometry b et we en their “con vex cores”, i.e. b et wee n closed conv ex subsets C ⊆ C S ( l ) and C ′ ⊆ C S ( l ′ ) whic h are minimal among the closed con v ex subsets satisfying ∂ ∞ C = ∂ ∞ C S ( l ) and ∂ ∞ C ′ = ∂ ∞ C S ( l ′ ). D ue to a basic rigidity phenomenon in the geometry of nonp ositiv ely curv ed spaces, the so-called “Flat Strip Theorem”, the con v ex cores are unique up to isometry . W e see that, esse ntially due to the connectedness of the Tits b oundary , there are many natural identiﬁcations b et w een the v ario us pa r allel sets and observ e that, b y comp osing them, one can generate large groups of isometries acting on the cores C l of the cross sec tio ns (see section 3). W e denote the closures o f these “holonom y” subgroups in I som ( C l ) b y H ol ( l ). They are lar g e in the sense that H ol ( l ) acts 2-fo ld transitiv ely on ∂ ∞ C S ( l ). 7 Hence, however unsymmetric X itself may b e, the cr oss se ctions of its p ar al lel sets ar e always highly symmetric, and this is the k ey observ ation at the starting p o in t of our argumen t. The high symme try imp oses a substantial restriction on the geometry o f the cross sections and the ma jor step in our pro of of 1.2 is a rank-1 analogue for spaces with high symmetry: Theorem 1.7 L et Y b e a lo c al ly c omp a c t Hadamar d sp ac e with extendi b le r ays and at le ast 3 p oints at inﬁnity. Assume that Y c ontain s a close d c onvex subset C wi th ful l ide al b oundary ∂ ∞ C = ∂ ∞ Y so that I som ( C ) acts 2-fo l d tr ansitively on ∂ ∞ C . Then the fol lowing dichotomy o c curs: 1. If some c omplete ge o desics in Y br anch then Y is isometric to the pr o duct of a metric tr e e (with e dges of e qual len gth) and a c omp act Hadamar d sp ac e . 2. If c omplete ge o desics in Y do not br anch then ther e exists a r a n k-1 Riemannian symmetric mo del sp ac e Y model , and a b oundary home om orphism ∂ ∞ Y → ∂ ∞ Y model c arrying I som o ( C ) to I som o ( Y model ) . 8 In particular, the ideal b oundary of ev ery cross section is homeomorphic to a sphere, a Can tor set or a ﬁnite set of cardinality ≥ 3. As w e explained, the geometry of X is r ig idiﬁed b y the v ario us iden tiﬁcatio ns b et we en cores of cross sections of para llel sets. This can b e nicely built in the picture of the geometric compactiﬁcation of X as follo ws: W e men tioned that the conv ex cores of the cross sections C S ( l ) for all lines l asymptotic to the same v ertex ξ ∈ ∂ T its X can b e canonically iden tiﬁed to a Hadamard space C ξ . It has rank 1 in the sense that it satisﬁes the visibilit y prop ert y , o r equiv alently , its Tits b oundary is discrete. ∂ ∞ C ξ can b e reinterprete d as the compact top olog ical space of W eyl c ham b ers (arcs) emanating from the verte x ξ . One can now blo w up the geometric b oundary ∂ ∞ X by replacing eac h v ertex ξ by the geometric compactiﬁcation ¯ C ξ and gluing the endpo ints of W eyl arcs to the corresp onding b oundary p oin ts in ∂ ∞ C ξ . This generalizes a construction 7 If X is a r ank-2 symmetric spa ce, H ol ( l ) contains the iden tity comp onent of the iso m- etry gr o up o f the rank-1 symmetric space C S ( l ). So in the example X = S L (3 , R ) /S O (3) ( X = S L (3 , C ) /S U (3)) the action of H ol ( l ) on the b o undary of the hyperb olic plane (h yp erb olic 3-space) C S ( l ) is even 3-fold tra nsitive (b y M¨ obius tra nsformations). Mor e ge ne r ally the a ction is 3-fold transitive if ∂ T its X is the s pherical building a s so ciated to an (abstract) pro jective plane. 8 It seems unclear whether in this ca se one should not be able to ﬁnd an embedded rank- 1 symmetric s pace inside Y . 6 b y Ka rp elevi ˇ c for sym metric spaces [Ka]. W e denote the resulting reﬁned b oundary by ∂ f ine ∞ X , and b y ∂ f ine ,∂ ∞ X the par t whic h one obtains b y inserting o nly the b oundaries ∂ ∞ C ξ instead of the full compactiﬁcations ¯ C ξ . The rigidity expr esses itself in the action of the holonomy gr oup oid which app e ars on the blown up lo cus of the r eﬁne d b oundary ∂ f ine ∞ X due to the c onne c te dness of ∂ T its X : F o r an y t w o an tip o da l v ertices ξ 1 , ξ 2 ∈ ∂ T its X , i.e. v ertices of Tits distance ∠ T its ( ξ 1 , ξ 2 ) = π , there is a canonical isometry C ξ 1 ↔ C ξ 2 (2) b ecause the spaces C ξ i em b ed as minimal conv ex subsets into the cross section C S ( { ξ 1 , ξ 2 } ) of the f a mily of parallel geo desics asymptotic to ξ 1 , ξ 2 . W e can com- p ose suc h isometries hopping a lo ng ﬁnite sequences o f succe ssiv e a ntipo des. F or any t w o v ertices ξ , η ∈ ∂ T its X we denote by H ol ( ξ , η ) ⊆ I som ( C ξ , C η ) the closure of the subset of all isometries C ξ → C η whic h arise as ﬁnite comp osites of isometries (2) (cf. section 3). In particular, the holono m y groups H ol ( ξ ) := H ol ( ξ , ξ ) act on the inserted spaces ¯ C ξ . These actions can be though t of as an additional geometric structure on the spaces ∂ ∞ C ξ , namely as the analogue of a conformal structure; for instance if ∂ ∞ C ξ is homeomorphic to a sphere then due to 1.7 it can b e iden tiﬁed with the b oundary of a rank-1 symmetric space up to conformal diﬀeomorphism. Comment on the pr o o f of 1.2: It is easy to see that all cross sections C S ( l ) ha v e extendible geo desic ray s if X is geo desically complete (5 .3). If some c omplete ge o de sics br anch in X then there is a cross section C S ( l ) with branc hing geo desics (5.6) and, apparently less trivially to v erify , eve n all cross sections ha v e this prop ert y (5.29). The rank-1 result 1.7 then implies that the cross sections of all parallel sets are metric trees (up to a compact factor ) . F rom this point it is fairly straigh t-fo rw ard to conclude in one wa y or another that X is a Euclidean building (section 5.4.3). If c omplete ge o desics in X do not br anch w e can adapt arguments of Gromo v fro m the pro of of his Rigidit y Theorem [BGS85]. The reﬂections at p o in ts x ∈ X giv e rise to in v olutiv e automorphisms ι x : ∂ ∞ X → ∂ ∞ X of the top ological spheric al building ∂ ∞ X . One obtains a prop er map X ֒ → Aut ( ∂ ∞ X ) in to the group of b oundary au- tomorphisms and hence ﬁnds oneself in the situation that the top ological spherical building ∂ T its X is highly symmetric. (It satisﬁes the so- called Moufang prop erty .) Aut ( ∂ ∞ X ) is a lo cally compact top ological group [BS87]. Similar to [BS87], estab- lishing transitivit y and con traction prop erties for the dynamics of Aut ( ∂ ∞ X ) o n ∂ ∞ X allo ws to sho w, using a dee p result b y Gleason and Y amab e on the approximation of lo cally compact top ological groups by Lie groups, that Aut ( ∂ ∞ X ) is a semisimp le Lie group and the isometry group of a Riemannian sym metric mo del space X model . The in v olutions ι x can be c hara cterized as or der 2 elemen ts with compact centralize r and hence corresp ond to p oint reﬂections in X model . One obtains a map Φ : X → X model whic h is clearly aﬃne in the sense that it preserv es ﬂats. It immediately follow s that Φ is a homothet y , concluding t he pro of of 1.2. Comment on the pr o of of 1.3: Any b oundary isomorphism (1) has con tinuous diﬀeren tials Σ ξ φ : Σ ξ ∂ T its X → Σ φξ ∂ T its X ′ (3) and hence lifts to a map ∂ f ine ,∂ ∞ X → ∂ f ine ,∂ ∞ X ′ (4) 7 of part ially reﬁned bo undaries. The diﬀeren tials (3) are confo rmal in the sense that they preserv e the holonom y action, i.e. the induced homeomorphisms H omeo ( ∂ ∞ C ξ , ∂ ∞ C η ) → H omeo ( ∂ ∞ C φξ , ∂ ∞ C φη ) carry the holonom y group oid H ol X to the ho lonom y gr oup oid H ol X ′ . This sets us on the trac k tow ards the pro of of 1.3: Aft er pro ving 1.2 w e ma y assume that X is a symmetric space or a Euclidean building. Then the C ξ are rank-1 symmetric spaces or metric trees, resp ectiv ely , and the diﬀeren tials Σ ξ φ actually extend to homotheties C ξ → C φξ . This means that the lift (4) of φ impro ves to a holonomy equiv aria nt map ∂ f ine ∞ X − → ∂ f ine ∞ X ′ (5) b et we en t he full reﬁned geometric b oundaries. Since p oints in the blow ups C ξ are equiv a lence classes of strongly asymptotic geo desics, (5) enco des a corresp ondence b et we en singular geo desics in X and X ′ . If X is a Euclidean building then this can b e used in a ﬁnal step to set up a correspo ndence b etw een v ertices whic h preserv es apartmen ts and extends to a homot hety X → X ′ , hence concluding the pro of of 1.3 in this case (section 5.5) . If X is a symmetric space then 1.3 already follo ws from the argumen ts in t he pro of of 1.2. The pap er is desorganized as follows: In section 2 w e discuss preliminaries. In particular w e establish the existence of con ve x cores for Hadamard spaces under fairly general conditions (section 2.1.2) a nd in tro duce the spaces of strong asymptote classes whic h will serv e as an imp ortan t to ol in the construction of the holonomy group oid. The holonom y group oid is discus sed in section 3 where we explain the sy mmetries of parallel sets. In section 4 we pro v e the rigidity results for “rank 1” spaces with high symmetry and in section 5 the main results for higher rank spaces. 2 Preliminaries 2.1 Hadamard spaces F or basics on Hadamard spaces and, more generally , spaces with curv ature b ounded ab ov e w e refer t o the ﬁrst tw o chapters of [Ba95] and section 2 of [KL96]. Spaces of directions and Tits b oundaries are discussed there and it is v eriﬁed that they are CA T(1) spaces. Let us emphasize that we mean b y the Tits b oundary ∂ T its X of the Hadamard space X the geometric b oundary ∂ ∞ X eq uipp ed with the Ti ts angle metric ∠ T its and not with the asso ciated path metric 9 . In the following paragraphs w e supply a few auxiliary facts needed later in the text. 2.1.1 Filling spheres at inﬁnit y b y ﬂats The follo wing result generalizes a n observ atio n b y Sc hro eder in the smoot h case, cf. [BGS85]. 9 If ∂ T its X is a spherical building then it ha s diameter π with re s pe ct to the path metric and hence the path metric coincides with ∠ T its . 8 Prop osition 2.1 L et X b e a lo c a l ly c om p act Hadamar d sp ac e and let s ⊆ ∂ T its X b e a unit spher e which do es not b ound a unit h e mispher e in ∂ T its X . Then ther e exis ts a ﬂat F ⊆ X with ∂ ∞ F = s . Pr o of: Let s b e isometric to the unit sphere of dimension d ≥ 0 and pic k d + 1 pairs of an tip o des ξ ± 0 , . . . , ξ ± d so that ∠ T its ( ξ ± i , ξ ± j ) = π / 2 and ∠ T its ( ξ ± i , ξ ∓ j ) = π / 2 (6) for all i 6 = j . If for some p oint x ∈ X a nd some index i holds ∠ x ( ξ + i , ξ − i ) = π then the union X ′ = P ( { ξ + i , ξ − i } ) of geo desics asymptotic to ξ ± i is non- empt y and s determines a ( d − 1)- sphere s ′ ⊆ ∂ T its X ′ whic h do es not b o und a unit hemisphere. Moreo v er any ﬂat F ′ ⊆ X ′ ﬁlling s ′ determines a ﬂat F ﬁlling s and w e are reduced to the same question with one dimension less. W e can hence pro ceed by induction on the dimension d and the claim follow s if w e can rule out the situation that ∠ x ( ξ + i , ξ − i ) < π (7) holds for all x and i . In t his case w e obtain a contradiction as follows . As sume ﬁrst that for some (and hence any ) p oin t x 0 ∈ X the inters ection o f the horoballs H b ( ξ ± i , x 0 ) is un b ounded and thus contains a complete geo desic ra y r . The ideal endp oin t η ∈ ∂ ∞ X of r satisﬁes ∠ T its ( η , ξ ± i ) ≤ π / 2 b ecause the Busemann func- tions B ξ ± i monotonically non- increase along r . By the triangle inequalit y follo ws ∠ T its ( η , ξ ± i ) = π / 2 b ecause ξ ± i are an tip o des. The CA T(1) prop erty of ∂ T its X then implies that there is a unit hemisphere h ⊆ ∂ T its X with cen ter η and b oundar y s , but this con tra dicts our assumption. Therefore the in tersection of the horoballs H b ( ξ ± i , x 0 ) is compact for a ll x 0 ∈ X and the con v ex f unction max B ξ ± i is prop er and assume s a minim um in some p oin t x . D enote b y r ± i : [0 , ∞ ) → X the ray with r ± i (0) = x and r ± i ( ∞ ) = ξ ± i . (6) implies that B ξ ± i non-increases along r ± j for i 6 = j . Hence, if x j denotes the midpo in t of the se g ment r + j (1) r − j (1) then B ξ ± i ( x j ) ≤ B ξ ± i ( x ) for all i and, b y (7), B ξ ± j ( x ′ j ) < B ξ ± j ( x ) for some p oint x ′ j ∈ xx j . This means that by replacing x we can decrease the v alues of one pair of Busem ann functions while not increasing the others. By iterating this pro cedure at most d + 1 times w e ﬁnd a p oin t x ′ with max B ξ ± i ( x ′ ) < max B ξ ± i ( x ), a con tradiction.  2.1.2 Con v ex cores F or a subset A ⊆ ∂ ∞ Y w e denote b y C A the family of closed conv ex subsets C ⊆ Y with ∂ ∞ C ⊇ A . C A is non-empt y , pa r t ia lly ordered and closed under in tersections. Prop osition 2.2 L et Y b e a lo c a l ly c om p act Hadamar d sp ac e. 1. Supp ose that s ⊆ A ⊆ ∂ ∞ Y and s is a unit s p her e with r esp e ct to the Tits metric which do es n o t b ound a unit hemispher e. Then C A c ontains a minimal element. 2. Supp ose that A ⊆ ∂ ∞ Y so that C A has minimal ele ments. Then the union Y 0 of al l mi n imal elem ents in C A is a c onvex subset of Y . It de c omp os es as a m etric pr o duct Y 0 ∼ = C × Z (8) wher e Z is a c omp act Hadamar d sp ac e and the layers C × { z } ar e pr e cisel y the minim a l elements in C A . 9 Pr o of: According t o 2.1, there exists a non-empt y family of ﬂats in Y with ideal b oundary s a nd the family is compact b ecause otherwise s would b ound a unit hemi- sphere. The union P ( s ) of these ﬂats is a con ve x subset of Y . Sublemma 2.3 L et F b e a ﬂat and C a close d c onvex subset in Y so that ∂ ∞ F ⊆ ∂ ∞ C . Then C c ontains a ﬂat F ′ p ar al lel to F . Pr o of: F or an y p oints x ∈ C and y ∈ F there is a p oin t x ′ ∈ C so that d ( x ′ , y ) ≤ d ( x, F ). Hence there exists a p oint x ′′ ∈ C whic h realizes the nearest po int distance of F and C : d ( x ′′ , F ) = d ( C , F ). Then the unio n of rays emanating from x ′′ and asymptotic to p oints in ∂ ∞ F fo rms a ﬂat F ′ parallel to F .  Hence every con vex subset C ∈ C A in tersects P ( s ) in a non- empt y compact family of ﬂats and therefore determines a non-empt y compact subs et U ( C ) in the compact cross section C S ( s ) (compare deﬁnition 3.4). W e o rder the sets C ∈ C A b y inclusion and observ e that the assignmen t C 7→ U ( C ) preserv es inclusion. Sublemma 2.4 L et ( S ι ) b e an or der e d de cr e asing famil y o f non-empty c omp act sub- sets of a c omp act m etric sp ac e Z . The n the interse ction of the S ι is not emp ty. Pr o of: F or ev ery n ∈ N we can co v er Z by ﬁnitely many balls of radius 1 /n and therefore there exists a ball B 1 /n ( z n ) whic h in tersects all sets S ι . Any accum ulat ion p oin t of the se quence ( z n ) is con tained in the interse ction of the S ι .  An y decreasing c ha in of sets C ι ∈ C A yields a decreasing c hain of compact cross sections U ( C ι ) and hence has non-empt y in tersection. It follow s that ∅ 6 = T C ι ∈ C A and, b y Z orn’s lemma or otherwise, we conclude that C A con tains a minimal non- empt y subset. No w let C 1 , C 2 ∈ C A b e minimal. F o r an y y 1 ∈ C 1 the closed con v ex subset { y ∈ C 1 : d ( y , C 2 ) ≤ d ( y 1 , C 2 ) } of C 1 con tains A in its ideal b oundary and, b y minimalit y of C 1 , is all of C 1 . It follo ws that d ( · , C 2 ) is constan t on C 1 and the nearest p oin t pro jection p C 2 C 1 : C 1 → C 2 is a n isometry . F or a decomp osition d ( C 1 , C 2 ) = d 1 + d 2 as a su m of p ositiv e n um b ers, the set { y ∈ Y : d ( y , C i ) = d i for i = 1 , 2 } is a minimal elemen t in C A . Hence Y 0 is con v ex. Sublemma 2.5 F or minimal elements C 1 , C 2 , C 3 ∈ C A the self-isometry ψ = p C 1 C 2 ◦ p C 2 C 3 ◦ p C 3 C 1 of C 1 is the iden tity. Pr o of: ψ preserv es the cen tral ﬂat f in C 1 with ideal b oundary s . F urthermore, ψ | f preserv es all Busemann functions cen tered at ideal p oints ∈ s . Th us ψ restricts to the iden tity on f . Since ∂ ∞ ψ = id and C 1 is minimal it fo llo ws that ψ ﬁxes C 1 p oin twise .  No w c ho ose a minimal set C ∈ C A and a p oin t y ∈ C . Then the set Z of p oints p C ′ C ( y ), where C ′ runs through all minimal elemen ts in C A , is conv ex. It is easy to see that Y 0 is canonically isometric to C × Z . Z m ust b e compact b ecause C S ( s ) is. This concludes the pro of of 2.2.  The compact Hadamard space Z in ( 8) has a w ell-deﬁned cen ter z 0 . W e call the la y er C × { z 0 } the c entr al minimal con vex subset in C A . 10 Deﬁnition 2.6 If C ∂ ∞ Y has min i m al elements then the con vex core cor e ( Y ) of Y is deﬁne d as the c entr al mi nimal close d c onvex subset in C ∂ ∞ Y . If the con v ex cor e exists it is pres erve d b y all isometrie s of Y . Lemma 2.7 L et Y b e a lo c al ly c omp act Hadamar d sp ac e which ha s a c onvex c or e. I f cor e ( Y ) h as no Euclide an factor then any isometry with trivial action at inﬁnity ﬁxes cor e ( Y ) p ointwise. Pr o of: Let φ b e an isometry whic h acts trivially at inﬁnity . Then its displacemen t function is constant on the cen tral conv ex subset C . It is zero b ecause C do es not split oﬀ a Euclidean factor.  2.1.3 Spaces of strong asymp t ote classes Let X b e a Hadamard space. F o r a p oint ξ ∈ ∂ ∞ X let us consider the ra ys asymptotic to ξ . The a symptotic d istanc e of tw o ra ys ρ i : [0 , ∞ ) → X is giv en b y their nearest p oin t distance d ξ ( ρ 1 , ρ 2 ) = inf t 1 ,t 2 →∞ d ( ρ 1 ( t 1 ) , ρ 2 ( t 2 )) , (9) whic h equals lim t →∞ d ( ρ 1 ( t ) , ρ 2 ( t )) when the ray s are parametrized so that B ξ ◦ ρ 1 ≡ B ξ ◦ ρ 2 . W e call the rays ρ i str ongly asymptotic if their asymptotic distance is zero. The asymptotic distance (9) deﬁnes a metric o n t he space X ∗ ξ of strong asymptote classes. Prop osition-Deﬁnition 2.8 The metric c ompletion X ξ of X ∗ ξ is a Hadam ar d sp a c e. Pr o of: Any t w o p oin ts in X ∗ ξ are represen t ed by ra ys ρ 1 , ρ 2 : [0 , ∞ ) → X asymptotic to ξ and initiating o n the same horosphere cen tered at ξ . Denote b y µ s : [ s, ∞ ) → X the ra y asymptotic to ξ whose starting p oin t µ s ( s ) is the midp oin t of ρ 1 ( s ) ρ 2 ( s ). The triangle inequalit y implies that d ( ρ 1 ( t ) , µ s ( t )) + d ( µ s ( t ) , ρ 2 ( t )) − d ( ρ 1 ( t ) , ρ 2 ( t )) ≤ d ( ρ 1 ( s ) , ρ 2 ( s )) − d ( ρ 1 ( t ) , ρ 2 ( t )) → 0 as s , t → ∞ with s ≤ t . Hence d ( µ s ( t ) , µ t ( t )) → 0 and d ξ ( µ s , µ t ) → 0, i.e. ( µ s ) is a Cauch y sequenc e and its limit in X ξ is a midp oint for [ ρ 1 ] and [ ρ 2 ]. In this manner w e can assign to ev ery pair of p oin ts [ ρ 1 ] , [ ρ 2 ] ∈ X ∗ ξ a w ell-deﬁned midp oin t m ∈ X ξ . If [ ρ ′ 1 ] , [ ρ ′ 2 ] ∈ X ∗ ξ is anot her pair o f p o in ts so that d ([ ρ i ] , [ ρ ′ i ]) ≤ δ then d ( m, m ′ ) ≤ δ . It follow s that t here exist midp o in ts for all pairs o f p oin ts in X ξ . As a conseque nce, an y tw o p oints in X ξ can b e connected by a geo desic. An y ﬁnite conﬁguration F of p oints in X ∗ ξ corresp onds to a ﬁnite set of r a ys ρ i : [0 , ∞ ) → X asymptotic to ξ and sync hronized so that for any time t the set F t of p oin ts ρ i ( t ) lies on one horosphere cen tered at ξ . The ﬁnite metric spaces ( F t , d X ) Hausdorﬀ con v erge to ( F , d ξ ) and hence distance comparison inequalities are inherited. It follows that geo desic triangles satisfy the CA T(0) comparison inequalit y .  W e will also X ξ call the sp a c e of str on g asymptote classes a t ξ ∈ ∂ ∞ X . It had b een considered by Ka r p elevi ˇ c in the case of symm etric spaces, see [Ka]. 11 2.1.4 T yp es of isometries W e recall t he standard classiﬁcation of isometries into axial, elliptic and parab olic ones: F o r any isometry φ of a Hadamard space X its displacemen t function δ φ : x 7→ d ( x, φx ) is con ve x. φ is called semisimple if δ φ attains its inﬁm um. The re are tw o t yp es of semisimple isometries: φ is el liptic if t he minim um is zero and has ﬁxed p oin ts in this case. If the minimu m is strictly p ositiv e then φ is axial and there is a non-empt y family of φ - in v ar ia n t parallel geo desics, the axes of φ . If δ φ do es not hav e a minim um then φ is called p ar a b olic . The ﬁxed p oin t set of a parab o lic isometry in ∂ T its X is non-empty and con tained in a closed ball of radius π / 2. Deﬁnition 2.9 F or ξ ∈ ∂ ∞ X we deﬁne the parab olic stabilizer P ξ as the gr oup c onsisting of al l el liptic and p ar ab olic i s ometries which pr es erve every hor ospher e c enter e d at ξ . Note that there are parab olic isometries whic h ﬁx more than one p oint at inﬁnity and do not preserv e the horospheres centere d at some of their ideal ﬁxed p oints . Deﬁnition 2.10 An isometry φ of a lo c al ly c omp act Hadamar d sp ac e X is c al le d purely parab olic iﬀ its c onjugacy class ac cumulates a t the identity. If I som ( X ) is c o c omp act then this i s e quivalent to the pr op e rty that for eve ry δ > 0 ther e exist arbitr arily lar ge b al ls on w hich the displac ement of φ is ≤ δ . 2.2 Visibilit y Hadamard spaces Let Y b e a lo cally compact Hadamard space with at least 3 ideal b oundar y p oin ts. W e assume that the Tits metric on ∂ ∞ Y is discrete, or equiv alently , that Y enjoys the visibility pr op erty in tro duced in [EO73]: a ny t wo p oints at inﬁnit y are ideal endp oints of some complete geo desic. Then an y t w o distinct ideal b oundary p oin ts ξ and η hav e Tits distance π and the family of (pa rallel) geo desics asymptotic to ξ , η is non-empt y and compact; w e denote their union by P ( { ξ , η } ). The visibilit y prop erty is clearly inherited by closed conv ex subsets. The terminology visibility is motiv at ed b y the follo wing basic fact: Lemma 2.11 F or e v ery y ∈ Y an d every ǫ > 0 ther e exists R > 0 such that the fol lowing is true: If pq is a ge o desic se gment not in terse cting the b a l l B R ( y ) then ∠ y ( p, q ) ≤ ǫ . Pr o of: See [EO73].  Consequence 2.12 L et A b e a c omp act subset of ∂ ∞ Y × ∂ ∞ Y \ D iag . Then the se t of al l ge o desics c ⊂ Y satisfying ( c ( −∞ ) , c ( ∞ )) ∈ A is c o mp act. Pr o of: This set B of geo desic is certainly closed. If B w ould contain an unbounded sequence of geodesics c n then the corresponding sequence of p oin ts ( c n ( −∞ ) , c n ( ∞ )) in A w o uld accum ulate at the diagonal ∆, con tradicting compactness.  12 Remark 2.13 Visibi li ty Hadamar d sp ac es with c o c omp act i s o metry g r oup ar e lar ge- sc ale hyp erb olic i n the sense of Gr omo v . A sequence ( φ n ) ⊂ P ξ div erges t o inﬁnit y , φ n → ∞ , iﬀ φ n con v erges to the constan t map with v alue ξ uniformly on compact subsets of ∂ ∞ Y \ { ξ } . Lemma 2.14 Assume that for diﬀer ent ide al p oints ξ , η ∈ ∂ ∞ Y ther e ar e se quenc e s of p ar ab olics φ n ∈ P ξ and ψ n ∈ P η diver gin g to inﬁnity. Then φ n ψ n is axial for lar ge n . Pr o of: Let U and V b e disjoin t neigh b orho o ds of ξ , η respectiv ely . Then φ ± 1 n ( ∂ ∞ Y \ U ) ⊂ U and ψ ± 1 n ( ∂ ∞ Y \ V ) ⊂ V for large n whic h implies α n ( ∂ ∞ Y \ V ) ⊂ U and α − 1 n ( ∂ ∞ Y \ U ) ⊂ V (10) with α n = φ n ψ n . α n can’t b e elliptic (for large n ) b ecause then ( ∂ ∞ α k n ) k ∈ N w ould sub con v erge to the iden tit y , con tradicting (1 0). α n can’t b e parab olic either b ecause then ( ∂ ∞ α k n ) k ∈ N w ould con ve rg e to a constant function eve ry where point wise, which is also excluded b y (10). Therefore α n is axial for large n .  2.3 Buildings: Deﬁnition, v o cabulary and examples A geometric treatment of spherical and Euclidean Tits buildings within the framew ork of Aleksandrov spaces with curv ature b ounded ab ov e has to some exten t b een carried through in [KL96]. W e will use these results and for the con v enience of the reader w e brieﬂy recall some of the basic deﬁnitions and concepts. 2.3.1 Spherical buildings A spheric al Coxeter c om p lex consists of a unit sphere S and a ﬁnite Weyl gr oup W ⊂ I som ( S ) generated b y reﬂections at wal ls , i.e. totally geo desic subspheres of co dimension 1. The w a lls divide S in to op en con v ex subsets whose closures are the chamb ers . These are fundamental domains f o r t he action of W on S and pro ject isometrically to the orbit space, the mo del W eyl c ham b er ∆ model = W \ S . A p anel is a co dimension-1 face of a c hamber. A spheric al building mo delled on the Co xeter complex ( S, W ) is a CA T(1) space 10 B together with an atla s of c harts, i.e. isometric em b eddings ι : S ֒ → B . The imag e of a chart is a n ap a rtme n t in B . W e require that any t w o p o ints are contained in an apartment and that the co ordinate changes b et wee n charts are induced b y isometries in W . The notions of wall, c ham b er, panel etc. transfer from t he Co xeter complex to the building. There is a canonical 1-Lipsc hitz contin uo us ac c or de on map θ B : B → ∆ model folding the building on to the model c ham b er so that ev ery c hamber pro jects isometrically . θ B ξ is called the typ e of a p oin t ξ ∈ B . ξ is r e gular if it lies in the in terior of a cham b er. B is thick if eve ry panel is adjacen t to at least 3 c han b ers. If B has no spherical de Rham factor, i.e. if W acts without ﬁxed p oints, then the c ham b ers a r e simplices and 10 A CA T(1) sp ac e is a complete geo des ic metric space with upper curv ature bound 1 in the sense of Aleksandrov. 13 B carries a natural structure of a piecew ise spherical simplicial complex . In this case w e’ll call the faces also simplices . A thic k spherical building B is called irr e d ucib le if the corresponding linear represen tation of W is irreducible. This is equiv alen t to the assertions that B do es not decompo se as a spherical join, a nd tha t ∆ model do es not decomp ose. Tits originally introduced buildings to inv ert F elix Kleins Erlanger Programm and to pro vide geometric in terpretatio ns for algebraic gro ups, i.e. to construct geometries whose auto morphism groups are closely related to these gr o ups. The simplest in- teresting examples of irreducible spherical buildings are the buildings asso ciat ed to pro jectiv e linear groups. In dimension 1, one can more generally construct a spherical building for ev ery a bstract pro j ectiv e plane, p ossibly with trivial group of pro jectiv e transformations: Example 2.15 Given an abstr act pr oje ctive plan e P one c onstructs the c orr esp ond- ing 1-dimensional irr e ducible spheric al building B ( P ) a s fol lows . T h e r e ar e two sorts of vertic e s in B ( P ) : r e d vertic es c orr es p ondin g to p oints in P and blue vertic es c orr e- sp ondi n g to lines. One dr a w s an e dge of length π / 3 b etwe en a r e d and a blue vertex iﬀ they ar e incident. The e dg es in B ( P ) c orr esp on d to lines in P with a marke d p oint. The ap artments in B ( P ) , i.e. clos e d p a ths of length 2 π and c onsisting of 6 e dges, c or- r esp ond to trip els of p oints (r es p e ctively lines) in gener al p osition. F r o m the incidenc e pr op erties of pr oje ctive planes one e asily de duc es that any two e dges ar e c ontaine d in an ap artment and that ther e ar e no close d p aths of length < 2 π , i.e. B ( P ) is a CA T(1) sp ac e. Of course, a top ological pro jectiv e plane yields a top ological spherical building. Remark 2.16 (Exotic smooth pro jective planes) As Bruc e Kleiner p ointe d out one c an pr o duc e exotic (smo oth) pr o je ctive planes by p erturbing a smo oth pr oje ctive plane, for instanc e one of the standa r d pr oje ctive planes P R 2 , P C 2 or P H 2 . 2.3.2 Euclidean buildin gs A Euclide an Coxeter c omplex consists of a Euclidean space E and an a ﬃne Weyl gr oup W af f ⊂ I som ( E ) generated b y reﬂections at wal ls , i.e. aﬃne subspaces o f co dimension 1, so that the imag e W of W af f in I som ( ∂ T its E ) is a ﬁnite reﬂection group and ( ∂ T its E , W ) th us a spherical Co xeter complex. A Euclid e an buildi n g is a Hadamard space X with the follo wing additional struc- ture: There is a canonical maximal atlas of isometric embeddings ι : E ֒ → X called charts so that the co ordinate changes are induced by isometries in W af f . Any geo desic segmen t , ra y and complete geo desic is contained in an ap artment , i.e. the image of a c hart. The c ha r t s assign to an y non-degenrate segmen t xy a we ll- deﬁned dir e ction θ ( xy ) in the anisotr opy p ol yhe dr on ∆ model , the mo del W eyl c ham b er of ( ∂ T its E , W ). W e request that for any tw o non-degenerate segmen ts xy and xz the angle ∠ x ( y , z ) tak es one of the ﬁnitely many v a lues whic h can o ccur in ( ∂ T its E , W ) as distance b e- t w een a p oint of t yp e θ ( xy ) and a po in t of t yp e θ ( xz ). ( This is called the angle rigi d ity pr op erty in [KL96].) The r ank of X is dim ( E ). The spaces of directions Σ x X a nd the Tits b oundary ∂ T its X inherite canonical spherical building structures mo delled on ( ∂ T its E , W ). X 14 is thick (irr e ducible) if ∂ T its X is thic k (irreducible). X is called discr ete if W af f is a discrete subgro up of I som ( E ). Thic k lo cally compact Euclidean buildings are discrete and they carry a natural structure as a piecewise Euclidean simplicial complex. Example 2.17 Euclide an build i n gs of dimension 1 ar e metric trees , i.e. sp ac es of inﬁnite ne gative curvatur e in the sense that al l ge o desic triangles de gener ate to trip o ds. Man y in teresting examples of lo cally compact irreducible Euclidean buildings a rise from simple algebraic groups ov er non-Arc himedean lo cally compact ﬁelds with a discrete v aluation. Example 2.18 L et K b e a lo c al ly c omp act ﬁeld with discr ete valuation, uniformizer ω , ring of inte gers O and r esidue ﬁeld k . The Euclide an building attache d to S L (3 , K ) is c onstructe d as fol lows: I t is a sim p l i c ial c omplex built fr om i s ometric e quilater al Euclide an triangles. The vertic es ar e p r oje ctive e quivalenc e classes of O -lattic es in the K -ve ctor sp a c e K 3 . Thr e e la ttic es Λ 0 , Λ 1 , Λ 2 r epr esent the vertic es of a triangle if, mo dulo r esc a ling and p ermutation, the inclusion ω · Λ 0 ⊂ Λ 1 ⊂ Λ 2 ⊂ Λ 0 holds. ∂ T its X is isomorphic to the spheric a l building attache d to the pr o j e ctive plane over K , and for any vertex v ∈ X the sp a c e of dir e ctions Σ v X is isomorphic to the spheric al building a ttache d to the pr oje c tive plane over the r esidue ﬁeld k . Remark 2.19 (Unsymmetric irreducible rank-2 Euclidean buildings) Ther e ar e diﬀer ent lo c a l ly c omp act ﬁelds with the same r esidue ﬁeld, an d h e n c e diﬀer ent buildings as in 2.18 with isometric sp ac es of dir e c tion s at their v e rtic es. In fact one c an c on struct unc ountably many building s such that the sp ac es of d ir e ctions at their vertic es ar e is o metric to the spheric al building attache d to a given pr oje ctive p lane. In this way one c an o b tain building s with no non-trivial symmetry and their b oundaries ar e spheric al buildings attache d to “exotic” top olo gic al pr oje c tive planes. 2.4 Lo c ally compact top ological groups W e will mak e essen tial use of a deep result due to Gleason a nd Y amab e on the appro ximation of lo cally compact top olog ical groups b y Lie groups: Theorem 2.20 (cf. [MZ55, p. 153]) Ev e ry lo c al ly c om p act top olo g i c al gr oup G has an op en sub gr oup G ′ such that G ′ c an b e appr oximate d by Lie gr oups i n the fol lowing sense: Every neigh b orho o d of the identity in G ′ c ontains an invariant sub gr oup H such that G ′ /H is isomorphic to a Lie gr oup. Here is a typical example of a non-Lie lo cally compact group: Let T b e a lo- cally ﬁnite simplicial tree and G it s isometry g roup eq uipp ed with the compact-op en top ology . V ertex stabilizers S tab ( v ) are op en compact subgroups homeomorphic to the Can tor set and can b e appro ximated b y ﬁnite groups; namely eve ry neigh b or- ho o d of the identit y in S tab ( v ) con ta ins the stabilizer of a ﬁnite set V of v ertices, v ∈ V ⊂ T , as no rmal subgroup of ﬁnite index. Other in teresting examples are pro vided by isometry groups of Euclidean and h yp erb olic buildings or more general classes of piecewise Riemannian complexes. 15 3 Holonom y Assumption 3.1 X is a lo c al ly c omp act Hadama r d sp ac e. ∂ T its X is a thick spheric al building o f dimension r − 1 ≥ 1 . F or a unit sphere s ⊂ ∂ T its X , 0 ≤ dim ( s ) < r − 1, w e denote by Link ( s ) the in tersection of all closed balls ¯ B π / 2 ( ξ ) cen tered at p oin ts ξ ∈ s . Link ( s ) is a closed con v ex subset and consists of the cen ters o f the unit hemispheres h ⊂ ∂ T its X with b oundary s . Not e tha t an y t w o of these hemis pheres in tersect precisely in s b ecause ∂ T its X is a CA T(1) space. It w on’t b e essen tia l for us but is w orth p ointing o ut that Link ( s ) carries a natural spherical building structure of dimension dim ( Link ( s )) = dim ( ∂ T its X ) − dim ( s ) − 1, compare Lemma 3.10 .1 in [KL9 6]. F or an y p o int ξ ∈ s w e hav e the natural map Link ( s ) → Σ ξ ∂ T its X (11) sending ζ to → ξ ζ . Both spaces Link ( s ) and Σ ξ ∂ T its X inherit a metric and a top ology from the Tits metric and cone top ology on ∂ T its X , and the injectiv e map (11) is a monomorphism in the sense that it preserv es b oth structures, i.e. it is contin uous and a Tits isometric em b edding 11 . Lemma 3.2 (11) maps Link ( s ) o n to Link (Σ ξ s ) . If dim ( s ) = 0 then Σ ξ s is empty and Link ( Σ ξ s ) is the full space of directions Σ ξ ∂ T its X . Pr o of: A direction → v ∈ Link (Σ ξ s ) corresp onds to a hemisphere h ⊂ Σ ξ ∂ T its X with b oundary Σ ξ s . Let ˆ ξ b e the an tip o de of ξ in s . Then the union of geo desics of length π with endp oints ξ , ˆ ξ and initial direc tio ns in h is a hemisphere whose cen ter ζ lies in Link ( s ) and maps to → v .  If s 1 , s 2 ⊂ ∂ T its X a r e unit spheres with dim ( s 1 ) = dim ( s 2 ) = dim ( s 1 ∩ s 2 ) ≥ 0 then fo r an y p oin t ξ in the interior of s 1 ∩ s 2 holds Σ ξ s 1 = Σ ξ s 2 = Σ ξ ( s 1 ∩ s 2 ) and the iden tiﬁcations Link ( s 1 ) → Link (Σ ξ ( s 1 ∩ s 2 )) ← Link ( s 2 ) yield an isomorphism Link ( s 1 ) ↔ Link ( s 2 ) (12) i.e. a cone top ology homeomorphism preserving the Tits metric. The followin g lemma sho ws that the identiﬁc at ion (12) do es no t depend on ξ : Lemma 3.3 F or the p o i n ts ζ i ∈ Link ( s i ) le t h i ⊂ ∂ T its X b e the unit hemispher es with c enter ζ i and b oundary s i . Then the p oints ζ i c orr esp on d to one another under (12) iﬀ the interiors o f the hemi s p her es h i have n o n-trivial in terse ction. 11 Recall tha t the topo logy induced by the Tits metric is ﬁner tha n the cone top o logy . 16 Pr o of: If the p o in ts ζ i corresp ond to o ne ano t her, i.e. → ξ ζ 1 = → ξ ζ 2 , t hen the segmen ts ξ ζ i initially coincide a nd the in teriors of the h i in tersect. Vice v ersa, if the interiors of the h i in tersect then for an y p oint ξ in the interior o f s 1 ∩ s 2 their in tersection h 1 ∩ h 2 is a neigh b orho o d of ξ in b oth closed hemisp heres ¯ h i and therefore → ξ ζ 1 = → ξ ζ 2 .  W e’ll now “ ﬁll in” the isomorphisms (12) by iden tiﬁcations of con ve x cores of cross sections of parallel sets in X . This will b e ac heiv ed by placing diﬀeren t cro ss sections into the same auxiliary am bien t Hadamard space, na mely a space of strong asymptote classes, so that their ideal b o undar ies coincide. Note that since X has spherical building b oundary , 2.2 implies that an y apartmen t a ⊂ ∂ T its X can b e ﬁlled b y a r - ﬂat F ⊂ X , i.e. ∂ ∞ F = a . If s ⊂ ∂ T its X is isometric to a unit sphere then s is con tained in an apartment (by [KL96 , Pro p osition 3 .9.1]) and he nce can b e ﬁlled b y a ﬂat f ⊂ X : ∂ ∞ f = s . This v eriﬁes that the parallel sets deﬁned next are non-empt y: Deﬁnition-Description 3.4 F or a unit spher e s ⊂ ∂ T its X we denote by P ( s ) = P X ( s ) the union of al l ﬂats with ide al b oundary s . P ( s ) is a n on-empty c on v ex subset and sp l i ts metric al ly as P ( s ) ∼ = R 1+dim s × C S ( s ) . (13) The subsets R 1+dim s × { point } a r e the ﬂats wi th id e al b oundary s . C S ( s ) is again a lo c al ly c omp act Hadamar d sp ac e which we c al l the cross section of P ( s ) . F or any ﬂat f ⊂ X , P ( f ) := P ( ∂ T its f ) denotes its parallel set , i.e. the union o f al l ﬂats p ar al lel to f , and C S ( f ) := C S ( ∂ T its f ) denotes the cr oss se ction. Observ e that ∂ T its C S ( s ) = Link ( s ). Namely a ray in C S ( s ) determines a ﬂa t half space in X whose ideal bo undar y is a hemisphere h in ∂ T its X with ∂ h = s ; vice v ersa, any su ch hemisphere in ∂ T its X can b e ﬁlled by a half- ﬂat in X . F or any point ξ ∈ s the natural map C S ( s ) → X ξ assigning to a p oint x the ra y xξ is an isometric em b edding b ecause for x 1 , x 2 ∈ C S ( s ) the triangle with vertice s x 1 , x 2 , ξ has right angles at the x i . Lemma 3.5 L et s 1 , s 2 ⊂ ∂ T its X b e unit s p her es with dim ( s 1 ) = dim ( s 2 ) = dim ( s 1 ∩ s 2 ) ≥ 0 . If ξ is a n interior p oint of s 1 ∩ s 2 then the images of the isometric emb e ddings C S ( s i ) ֒ → X ξ (14) have the same ide al b oundary. F urthermor e the r esulting identiﬁc ation of ide al b ound- aries c oincides with the e a rli e r identiﬁc ation (12). Pr o of: Let ζ i ∈ ∂ T its C S ( s i ) = Link ( s i ) be p oints corresp onding to eac h other under (12), i.e. → ξ ζ 1 = → ξ ζ 2 . The segmen ts ξ ζ i initially coincide, i.e. they share a non-degenerate segmen t ξ η . Let r i b e a ray in C S ( s i ) asymptotic to ζ i and r ′ i ⊂ C S ( s i ) b e the ray with same initial p oint but asymptotic to η . Then r i and r ′ i ha v e the same image in X ξ under (14) b ecause they lie in a ﬂat half- pla ne whose b oundary geo desic is asymptotic to ξ . Since the ra ys r ′ 1 and r ′ 2 are asymptotic this shows that the images of r 1 and r 2 in X ξ are asymptotic ra ys.  The Tits b oundaries ∂ T its C S ( s ) = Link ( s ) contain top-dimensional unit spheres and 2.2 implies that the cross sections C S ( s ) ha v e a conv ex core. 17 Lemma 3.6 I f s ⊂ ∂ T its X is a singular spher e then Link ( s ) do es not splitt oﬀ a spheric a l join factor. As a c onse quenc e, the c onvex c or e of C S ( s ) has no Euclide an factor. Pr o of: If Link ( s ) w o uld hav e a spherical join fa ctor then this factor w ould b e con- tained in all maximal unit spheres in Link ( s ). Hence the in tersection of all apartmen ts a ⊂ ∂ T its X with a ⊃ s w ould con tain a larger sphere t ha n s . This is imp ossible b e- cause ∂ T its X is a thic k spherical building and the singular sphere s is therefore an in tersection of apartmen t s.  Fix a simplex τ ⊂ ∂ T its X and c ho ose a p oint ξ in the interior of τ . Then the cross sections C S ( s ) for all singular spheres s ⊃ τ with d im ( s ) = dim ( τ ) isometrically em b ed in to the same am bient Hadamard space X ξ . By 3.5 their images hav e equal ideal b oundaries and the b o undary iden tiﬁcation is giv en b y (12). According to the pro of o f part 2 of 2 .2, the con v ex cores of the C S ( s ) are mapp ed to pa r a llel lay ers of a ﬂat strip and their b oundary identiﬁcations (12) can b e induced b y isometries whic h are uniq ue in view of 2.7 and 3.6. In this w a y w e can compatibly iden tify the con ve x cores in consideration to a Hadamard space C τ and there is a canonical isomorphism ∂ T its C τ ∼ = − → Σ τ ∂ T its X . (15) If σ, τ are top-dimensional simplices in the same singular sphere s ⊂ ∂ T its X then there is a canonical p ersp ectivit y isometry per sp στ : C σ ↔ C τ : pers p τ σ (16) b ecause b o t h sets are iden tiﬁed with the con ve x core of C S ( s ). The map of ideal b oundaries induced by (16) turns via (15) in to an isomorphism Σ σ ∂ T its X ↔ Σ τ ∂ T its X (17) (of top ological buildings) whic h can b e describ ed ins ide the Tits b oundary as follows: → u ∈ Σ σ ∂ T its X and → v ∈ Σ τ ∂ T its X corresp ond to eac h other if they are tangen t to the same hemisphere in ∂ T its X with b oundary s . ( 17) is indep enden t of the c hoice of s ⊃ σ ∪ τ . Let τ , ˜ τ ⊂ ∂ T its X b e simplices o f equal dimension a nd supp ose that they are pr oje ctively e q uiva l e nt , i.e. there exists a sequence τ = τ 0 , . . . , τ m = ˜ τ of simplices of the same dimension so that an y t w o succes siv e simplices τ i , τ i +1 are top- dimensional simplices in a singular sphere. By comp osing the natura l isometries (16) , C τ i → C τ i +1 , w e obtain an isometry C τ → C ˜ τ (18) Deﬁnition 3.7 The top olo gic al sp ac e H ol X ( τ , ˜ τ ) ⊆ I som ( C τ , C ˜ τ ) is deﬁne d as the closur e of the subset of iso m etries (18 ) . T he holonom y group H ol ( τ ) = H ol X ( τ ) ⊆ I som ( C τ ) at the simplex τ is deﬁne d as the top olo gic al gr oup H o l X ( τ , τ ) . 18 F or a face τ ⊂ ∂ T its X w e’d no w lik e to relate the holono my group oid on the space C τ to the holonom y group o id o n X . This will be useful in the pro of of 3.8 b ecause it allo ws to reduce the study of the holo nom y action to the rank 2 case. Let s, S ⊂ ∂ T its X b e unit spheres so that s ⊂ S . Let us denote b y s ⊥ ⊂ S the subsphere comple mentary to s , i.e. s ⊥ = Link S ( s ) and S = s ◦ s ⊥ . There are na tural inclusions Link ( S ) ⊂ Link ( s ) and P ( S ) ⊂ P ( s ). More precisely holds Link ( S ) ∼ = Link Link ( s ) ( s ⊥ ) (19) and C S ( S ) ∼ = C S C S ( s ) ( s ⊥ ) . (20) Assume now that the spheres s, S are singular and that τ ⊂ s and T ⊂ S are top-dimensional simplices in these spheres so that τ is a fa ce of T . The iden tiﬁ- cation Link ( s ) ∼ = Σ τ ∂ T its X carries s ⊥ to Σ τ S and Link Link ( s ) ( s ⊥ ) to Link ( Σ τ S ). cor e ( C S ( s )) ∼ = C τ carries cor e ( C S ( S )) ∼ = C T to cor e ( C S C τ (Σ τ S )) ∼ = C C τ Σ τ T and hence induces a canonical iden tiﬁcation C T ∼ = − → C C τ Σ τ T . (21) Tw o faces T 1 , T 2 ⊃ τ are t o p-dimensional simplic es in the same singular sphere S iﬀ the Σ τ T i are top-dimensional simplices in the same singular sphere in Σ τ ∂ T its X . L et us assume t ha t this we re the case. Then the p ersp ectivit y C T 1 ↔ C T 2 induces the p ersp ectivit y C C τ Σ τ T 1 ↔ C C τ Σ τ T 2 . W e obta in an em b edding H ol C τ (Σ τ T ) ֒ → H ol X ( T ) . (22) W e come to the main result of this section, namely that in the irreducible case the holonom y groups are non- trivial, ev en lar g e: Prop osition 3.8 Supp ose that, in addition to 3.1, the spheric al buildin g ∂ T its X is irr e ducible of dim ension ≥ 1 . T hen for any p anel τ ⊂ ∂ T its X a nd any η ∈ ∂ ∞ C τ , the p ar ab oli c stabilize r P η in H ol ( τ ) acts tr an sitively on ∂ ∞ C τ \ { η } . Pr o of: Let us ﬁrst consider the case dim ∂ T its X = 1. The panel τ is then a vertex ξ . The action of H ol ( ξ ) at inﬁnit y on ∂ ∞ C ξ ∼ = Σ ξ ∂ T its X can b e ana lysed inside ∂ T its X : Sublemma 3.9 F or every vertex ξ , H ol ( ξ ) acts 2 -fold tr ansitively on Σ ξ ∂ T its X . Pr o of: D enote b y l the length of W eyl c hambers. Irreducibility implies l ≤ π / 3. Consider t wo v ertices ξ 1 and ξ 2 of distance 2 l and let µ b e the midp oint of ξ 1 ξ 2 . Extend ξ 1 µξ 2 in an arbitra ry w ay to a (not nec essarily glo bally minimizing) geo desic η 1 ξ 1 µξ 2 η 2 of length 4 l . By irreducibilit y , this geo desic is contained in an apartmen t α for a ny choice of η 1 and η 2 . Denote b y ˆ µ the an tip o de of µ in α and let ζ 6∈ α b e some neigh b oring v ertex of µ . Then ∠ T its ( ζ , ξ i ) = π and w e can form the compo sition of natural maps (17): Σ ξ 1 ∂ T its X → Σ ζ ∂ T its X → Σ ξ 2 ∂ T its X . 19 V arying η 1 , η 2 , ζ we get plent y of maps Σ ξ 1 ∂ T its X → Σ ξ 2 ∂ T its X sending → ξ 1 ξ 2 = → ξ 1 µ to → ξ 2 ξ 1 = → ξ 2 µ and → ξ 1 η 1 to → ξ 2 η 2 . W e can comp ose these and their inv erses to o btain selfmaps of Σ ξ 1 ∂ T its X and see that the stabilizer of → ξ 1 ξ 2 in H ol ( ξ 1 ) acts transitiv ely on the complemen t of → ξ 1 ξ 2 . Since ∂ T its X is thic k, Σ ξ ∂ T its X con tains at least three p oin ts and it f ollo ws tha t H ol ( ξ ) acts 2-fold transitiv ely .  Pr o of of 3.8 c o ntinue d: If P η do es not act transitive ly on ∂ ∞ C τ \ { η } then, b y 3.9, there is a non- trivial axial isometry α ∈ H ol ( ξ ) (ﬁxing η ), a nd for an y ζ ∈ ∂ ∞ C τ \ { η } there is a conjugate α ζ of α with at t r activ e ﬁxed point η and repulsiv e ﬁxed p oin t ζ . F or ζ 1 , ζ 2 6 = η the isometries α − n ζ 2 ◦ α n ζ 1 ∈ P η sub con v erge to β ∈ P η with β ζ 1 = ζ 2 . This concludes the pro of in the 1-dimensional case. The gene ra l case dim ∂ T its X ≥ 1 can be deriv ed: Thanks to irreducibilit y , w e can ﬁnd for eve ry panel τ an adjacent panel ˆ τ so that µ := τ ∩ ˆ τ has co dimens ion 2 and ∠ µ ( τ , ˆ τ ) < π / 2. The building ∂ T its C µ ∼ = Σ µ ∂ T its X is 1- dimensional irreducible. Σ µ τ is a ve rtex and H ol C µ (Σ µ τ ) acts b y isometries on C C µ Σ µ τ ∼ = C τ . W e get an em b edding H ol C µ (Σ µ τ ) ֒ → H ol X ( τ ) as in (22). Our result in the 1-dimensional case implies t he assertion.  Example 3.10 If ∂ T its X is the spheric al building asso ciate d to a pr oje ctive plan e (with mor e than thr e e p oints) then H ol ( ξ ) acts 3-fold tr ansitive on Σ ξ ∂ T its X (by “M¨ obius tr ansfo rmations”). 4 Rank one : Rigidit y of high ly symmetric visibil- it y s p aces Assumption 4.1 L et Y b e a lo c al ly c omp act Hada m ar d s p ac e w ith at le ast thr e e ide al b ound a ry p oi n ts, with ex tend i ble r ays, and which is minim al in the sense that Y = co r e ( Y ) . Supp o s e furthermor e that H ⊆ I som ( Y ) is a close d sub gr oup so that for e ach ide al b ounda ry p oint ξ ∈ ∂ ∞ Y the p ar ab olic stabilizer P ξ in H acts tr ansitively on ∂ ∞ Y \ { ξ } . In particular, Y has the visibilit y prop erty . F or an y complete geo desic c w e denote b y P ( c ) the p ar al lel set of c , that is, the union of all geo desics parallel to c . It splits as c × cpt and contains a distinguished c entr al geo desic. By 2.14 there exist axial elemen ts in H , and hence the stabilizer of any cen tral geo desic con t a ins axial elemen ts. In particular, H a cts cocompactly on Y and Y is large-scale h yp erb olic (in the sense of Gromo v). F or an y orien ted cen tral g eo desic c there is a canonical homomorphism tr ans : S tab ( c ) → R (23) giv en b y the translational part. Its image is non-trivial closed, so either inﬁnite cyclic or R . The main result of this section is: Theorem 4.2 1. (23) is surje ctive iﬀ c omplete ge o desics in Y do not br anch. In this c ase, H is a simple Lie gr oup, ther e exists a ne gatively curve d symmetric sp ac e Y model and a home omorphism β : ∂ ∞ Y ∼ = − → ∂ ∞ Y model 20 which c arries H o to I som o ( Y model ) : β H β − 1 = I so m o ( Y model ) ⊂ H omeo ( ∂ ∞ Y model ) . 2. The image of (23 ) is cyclic iﬀ Y splits metric al ly as tr ee × cpt . 3. If T 1 , T 2 ar e two ge o desic al ly c o mplete lo c a l ly c omp act metric tr e es (w ith a t le ast thr e e ide al b oundary p oints), an d if ther e ar e e m b e ddings of top olo gic al gr oups H ֒ → I som ( T i ) satisfying 4 . 1, then ther e is an H -e quivaria n t homothety T 1 → T 2 . 4.2 is a com bination of the results 4.24 , 4.15 and 4.20. 4.1 General prop erties Lemma 4.3 L et ρ : [0 , ∞ ) → Y b e a r ay asymptotic to the g e o desic c . Then ρ is str ongly asymptotic to P ( c ) , i.e. d ( ρ ( t ) , P ( c )) → 0 . Pr o of: Assume that ρ has strictly p o sitive distance d from P ( c ). The stabilizer of P ( c ) con tains axial elemen ts with repulsiv e ﬁxed p oin t ρ ( ∞ ). Applying them to ρ w e can construct a geo desic a t p ositiv e distance from P ( c ), contradicting the deﬁnition of parallel set.  Lemma 4.4 L et c b e a ge o desic an d B ± Busemann functions c enter e d at the id e al endp oi n ts c ( ±∞ ) . Then the set wher e the 2-Lipsch itz func tion B + + B − attains its minimum is pr e cisely P ( c ) . Pr o of: Clear.  Lemma 4.5 F or every h > 0 ther e exists α = α ( h ) < π so that the fol lowing impli- c ation holds: If c : R → Y is a ge o desic, y a p oin t with ∠ y ( c ( −∞ ) , c (+ ∞ )) ≥ α then d ( y , P ( c )) ≤ h . Pr o of: Supp ose that for some p ositiv e h there is no α < π with this prop ert y . Then there exist p oin ts y n of distance ≥ h from P ( c ) so that α n = ∠ y n ( c ( −∞ , + ∞ )) → π . (All cen tral geo desics are equiv alent mo dulo the action of H .) This implies that there exist p oints y ′ n (on y n π P ( c ) ( y n )) so that d ( y ′ n , P ( c )) = h and ∠ y ′ n ( c ( ±∞ ) , π P ( c ) ( y n )) → π / 2 12 . Since H acts co compactly , w e ma y assume that the y ′ n sub con v erge. T aking a limit, w e can construct a geo desic parallel to c and at p o sitiv e distance h from P ( c ), a con tradiction.  4.2 Butterﬂy construction of small axial isometries Consider t w o ra ys ρ i : [0 , ∞ ) → Y emanating fro m the same point y and a ssume that ∠ y ( ρ 1 , ρ 2 ) < π . Let c i : R → Y b e extensions of the ra ys ρ i to complete g eo desics. W e pro duce an isometry ψ pr eserving the parallel set P ( c 1 ) b y comp osing four para b olic isometries: Let p i, ± ∈ P ( c i ( ±∞ )) b e the isometry whic h mo v es c i ( ∓∞ ) to c 3 − i ( ∓∞ ). Then ψ := p − 1 1 , + p 2 , − p − 1 2 , + p 1 , − 12 F or a closed co nv ex subset C of a Hadamard spac e X , π C : X → C denotes the closest p oint pro jection. 21 preserv es P ( c 1 ) and translates it b y the displacemen t δ ψ =  X B i, ± ( y )  − min( B 1 , + + B 2 , − ) − min ( B 1 , − + B 2 , + ) ≥ 0 to w a r ds c 1 (+ ∞ ). The displacemen t δ ψ is p ositiv e and ψ axial iﬀ one of the angles ∠ y ( c 1 ( ±∞ ) , c 2 ( ∓∞ )) is smaller than π . On the o ther hand, δ ψ is b ounded from a b ov e b y t wice the sum of the distance s from y to the parallel sets Y ( c 1 ( ±∞ ) , c 2 ( ∓∞ )). Lemma 4.6 I f ∠ y ( ρ 1 , ρ 2 ) ≤ π − α ( h ) then δ ψ ≤ 4 h . Pr o of: Since ∠ y ( c 1 ( ±∞ ) , c 2 ( ∓∞ )) ≥ α ( h ), 4.5 implies d ( y , P ( c 1 ( ±∞ ) , c 2 ( ∓∞ ))) ≤ h . Hence B 1 , ± ( y ) + B 2 , ∓ ( y ) − min( B 1 , ± + B 2 , ∓ ) ≤ 2 h and the claim follows .  4.3 The discrete case Assumption 4.7 (23) has cyclic image: T he stabilizer in H of any c en tr al ge o de s ic has a discr ete orbi t on the c entr al ge o desic. Then t here is a po sitiv e lo w er b ound for the displacemen t of axial isometries in H . By 4.6 there exists α 0 > 0 suc h that: If the ra ys ρ 1 and ρ 2 initiate in t he same p oin t y and ha v e angle ∠ y ( ρ 1 , ρ 2 ) < α 0 then ρ i ( ∞ ) hav e the same y -an tip o des (i.e. for a third ra y initiating in y we ha v e ∠ y ( ρ, ρ 1 ) = π iﬀ ∠ y ( ρ, ρ 2 ) = π ). Lemma 4.8 (No small angles betw een ra ys) I f the r ays ρ 1 and ρ 2 initiate in the same p oint y and ha ve angle < α 0 then they initial ly c oincide, i.e. ρ 1 ( t ) = ρ 2 ( t ) for smal l p ositive t . Pr o of: F o r small p o sitiv e t holds ∠ ρ 1 ( t ) ( ρ 1 ( ∞ ) , ρ 2 ( ∞ )) < α 0 , so ρ i ( ∞ ) hav e the same ρ ( t )-an tip o des 13 and ρ 2 ( t ) = ρ 1 ( t ).  Lemma 4.9 (Bounded Diving Time) If ρ : [0 , ∞ ) → Y is a r ay a symptotic to c and if d ( ρ (0) , P ( c )) ≤ h then ρ ( t ) ∈ P ( c ) for al l t ≥ h/ sin( α 0 ) . Pr o of: W e extend ρ to a geo desic c ′ . ρ is strongly asymptotic to P ( c ) (4.3). Hence there exist y n ∈ P ( c ) tending to ρ ( ∞ ) so that the rays ρ n = y n c ′ ( −∞ ) Hausdorﬀ con v erge to c ′ . ∠ y n ( c ( −∞ ) , c ′ ( −∞ )) → 0 and ρ n therefore initially lies in P ( c ) for large n (4.8). Outside P ( c ) the deriv ativ e of d ( ρ n ( t ) , P ( c )) is ≤ − sin( α 0 ) whence the estimate.  Corollary 4.10 (Discrete B ranc hing) Ther e exist br anching c omplete ge o desics : A ny two str o n gly asymptotic ge o d esics sh a r e a r ay. F urthermor e, the set of br a n ching p oints on any ge o de sic c is discr ete. 13 Let x b e a p oint in the Hadamard spa ce X . Then ξ , η ∈ ∂ ∞ X are x - antip o dal to eac h other if there ex is ts a geo des ic passing through x and asymptotic to ξ , η . 22 Pr o of: The ﬁrst assertion is clear from 4.9. The second follo ws from lo cal compactness: Let c n b e a sequence of g eo desics so that c n ∩ c = c n (( −∞ , 0]) and the bra nching p oin ts c n (0) are pairwise distinct a nd con ve rg e. Then, for large n , the p oin ts c n (1) are uniformly se para t ed (b y 4.8) but they form a bounded subset, con tradiction.  Prop osition 4.11 (Lo cal Conicalit y) L et ρ : R + → Y b e a ge o desic r ay, σ : [0 , l ] → Y a se g m ent so that ρ (0) = σ (0) . Then ther e exists t 0 > 0 so that the triangle with vertic es σ (0 ) , σ ( t 0 ) , ρ ( ∞ ) sp ans a ﬂat half-strip and is c ontaine d in a ﬂat strip. Pr o of: Denote b y ρ t : R + → Y the ray emanating from σ ( t ) and asymptotic to ρ . ρ t can be extended to a geo desic c t and there is a para llel geo desic c ′ t strongly asymptotic to ρ . The branch point of c ′ t and ρ tends to ρ (0) as t → 0. Discreteness of branc hing p oin ts on geo desics (and hence rays) implies that c ′ ( t ) passes through ρ (0) for small t , and σ | [0 ,t ] lies in the ﬂat strip b ounded b y c t and c ′ t .  Consequence 4.12 L et ρ 1 , ρ 2 : R + → Y b e r a ys emanating fr om the same p oint y and with angle ∠ y ( ρ 1 , ρ 2 ) = α . Then ρ 1 c an b e extende d to a c omplete ge o desi c c 1 such that ∠ y ( ρ 2 ( ∞ ) , c 1 ( −∞ )) = π − α . Consequence 4.13 (F attening half-strips) L et η ∈ ∂ ∞ Y and supp ose that σ : [0 , b ] → Y , 0 < b , is a se gment which is c ontaine d in a c omplete ge o desic (r ay). Assume that the ide al triangle ∆( σ (0) , σ ( b ) , η ) b ounds a ﬂat half-strip. Then we c an extend the se gmen t σ to a longer se gment σ : [ a, b ] → Y , a < 0 , so that the ide al triangle ∆( σ ( a ) , σ ( b ) , η ) b ounds a ﬂat half-strip. Pr o of: W e assume 0 < ∠ σ (0) ( σ ( b ) , η ) < π b ecause otherwise the claim ho lds trivially . Let ρ : R + → Y b e a ra y extending σ , i.e. ρ | [0 ,b ] ≡ σ . By 4.12 , we can ﬁnd a geo desic c extending ρ and a ﬂat strip S b ounded b y c so that the ra y σ (0) η is initially con tained in S . Then ∠ σ (0) ( c ( −∞ ) , η ) + ∠ σ (0) ( c (+ ∞ ) , η ) = π . F or a < 0 suﬃcien tly close to 0 the ideal triangle ∆( c ( a ) , c (0) = σ (0) , η ) b ounds a ﬂat ha lf -strip, hence ∠ c ( a ) ( c ( b ) , η ) + ∠ c ( b ) ( c ( a ) , η ) = π and ∆( c ( a ) , c ( b ) , η ) b ounds a ﬂat half-strip.  Corollary 4.14 T he angl e b etwe en any two r a ys emanating fr om the same p oint is 0 or π . Pr o of: Supp o se that ρ 1 , ρ 2 : R + → Y are t wo rays emanating from the same p oin t y with angle ∠ y ( ρ 1 , ρ 2 ) = α . F or small t , the ideal triangle ∆( ρ 1 (0) , ρ 1 ( t ) , ρ 2 ( ∞ )) b ounds a ﬂat half -strip (4.11). By 4.13 a nd lo cal compactness w e can extend ρ 1 to a complete geo desic c 1 : R → Y so that ∠ c 1 ( − t ) ( ρ 1 ( ∞ ) , ρ 2 ( ∞ )) = α for all − t ≤ 0. Since Y is large-scale hy p erbolic this implies that α = 0 or π .  Prop osition 4.15 Y splits as tr ee × compact . Pr o of: According to 4.14, for ev ery y ∈ Y the union S un y of all ra ys initiating in y is a minimal closed conv ex subse t isometric to a metric tree. 2.2 implies the assertion.  23 Prop osition 4.16 L et Y ′ b e a lo c al ly c omp a c t Hada m ar d sp ac e with ex tendible r ays and supp ose that T = cor e ( Y ′ ) exists and is a metric tr e e. Then Y ′ ∼ = T × cpt . Pr o of: The tree T is lo cally compact and geodesically complete, so it is also discrete. Sublemma 4.17 T he ne ar est p oint pr oje ction π T : Y ′ → T r estricts to an is o metry on eve ry r ay r in Y ′ . Pr o of: W e can extend r to a complete geo desic l and observ e that the distance d ( · , T ) from T is constan t on l b ecause l ( ±∞ ) ∈ ∂ ∞ T . It follo ws t hat π T restricts on l to an isometry .  Sublemma 4.18 L et y ∈ Y ′ and ξ 1 , ξ 2 ∈ ∂ ∞ Y ′ so that ∠ π T y ( ξ 1 , ξ 2 ) = π . Then ∠ y ( ξ 1 , ξ 2 ) = π . Pr o of: F or p oints y i on the ra ys y ξ i w e hav e d ( y 1 , y 2 ) ≥ d ( π T y 1 , π T y 2 ) = d ( π T y 1 , π T y ) + d ( π T y , π T y 2 ) = d ( y 1 , y ) + d ( y , y 2 ) ≥ d ( y 1 , y 2 ) . Th us equalit y holds and ∠ y ( y 1 , y 2 ) = π .  Sublemma 4.19 L et ξ ∈ ∂ ∞ Y ′ and c b e a ge o des i c in Y ′ not asymptotic to ξ . Th en ther e is a p oint y ∈ c with ∠ y ( l ( ±∞ ) , ξ ) = π . Pr o of: Let y b e the p oint whic h pro jects via π T to the cen ter of the trip o d in T spanned b y the ideal p o ints l ( ±∞ ) , ξ and apply 4.18.  Th us a n y tw o ra ys in Y ′ with same initial p oin t hav e angle 0 or π and 4.16 f ollo ws.  4.3.1 Equiv arian t rigidit y for trees Supp ose that T 1 and T 2 are geo desically complete lo cally compact metric trees with at least three b o undar y p oin ts, that the lo cally compact top ological gro up H is em- b edded in to their isometry groups, H ⊆ I som ( T i ), and that the induced b oundary actions of H on ∂ ∞ T i satisfy 4.1. Prop osition 4.20 Every H -e quivariant home omorphism ∂ ∞ T 1 → ∂ ∞ T 2 is induc e d by an H -e quivariant homothety T 1 → T 2 . Pr o of: Maximal compact subgroups K ⊂ H whose ﬁxed p oin t set o n T i is a v ertex (and not the midp o in t of a n edge) can b e recognize d from their dynamics at inﬁnit y: There exist three ideal b oundary p oin ts so that one can map a ny one to an y o ther of them b y isometries in K while ﬁxing the third. Adja cency of v ertices can b e c haracterised in terms of stabilizers: The v ertices v , v ′ ∈ T i are adjacen t iﬀ S tab ( v ) ∩ S tab ( v ′ ) is con ta ined in precisely tw o maximal compact v ertex stabilizers. It follo ws that there is a H -equiv a r ia n t com binatorial isomorphism T 1 → T 2 . It is a ho mothet y b ecause all edges in T i ha v e equal length.  24 4.4 The non-discr ete case Assumption 4.21 (23) i s surje ctive: Th e stabilizer in H of a n y c entr a l ge o desi c c acts tr a n sitively on c . Lemma 4.22 L et G b e an op en sub gr oup of H and c a c entr al ge o desic. Then S tab G ( c ) acts tr ansitive l y on c . Pr o of: W e c ho ose elemen ts h n ∈ S tab H ( c ) with tr ans ( h n ) = 1 / n . They form a b ounded seque nce and subconv erge to an elliptic elemen t k ∈ F ix H ( c ). Then ( k − 1 h n ) sub con v erges to e and there exist arbitra rily la rge m 6 = n so tha t h − 1 m h n is a xial and con tained in G . This shows that S tab G ( c ) contains axial elemen ts with arbitrar ily small non-v anishing translationa l part.  Consequence 4.23 Any op en sub gr oup of H acts c o c omp actly on Y . Prop osition 4.24 Ther e exi s t a ne gatively curve d s ymmetric sp ac e Y model , an iso- morphism H o ∼ = → I som o ( Y model ) and an e quivariant ho m e omo rp h ism ∂ ∞ Y → ∂ ∞ Y model . Pr o of: Supp ose that G ′ ⊆ H is an o p en subgroup and that K is an in v ariant compact subgroup of G ′ . 4.22 show s that the G ′ -in v a rian t non-empt y closed con v ex subset F ix ( K ) has full b oundar y at inﬁnit y: ∂ ∞ F ix ( K ) = ∂ ∞ Y . The minimalit y of Y implies F ix ( K ) = Y and K = { e } . Applying 2.20 we conclude that H is a Lie group. Sublemma 4.25 H has no non-trivial invariant a b elian sub gr oup A . Pr o of: A w ould hav e a non-empt y ﬁxed p oin t set in t he geometric compactiﬁcation Y . If A ﬁxes points in Y itself t hen F ix ( A ) = Y and A = { e } b y the co compactness of H and the minimality of Y . If all ﬁxed p oints of A lie at inﬁnity then there are at most t w o. This leads to a con t r a diction b ecause the ﬁxed p oin t set of A on ∂ ∞ Y is H -in v aria n t, hence full or empt y .  So H is a semisimple Lie group with trivial cen ter and H o ∼ = I som o ( Y model ) for a symmetric space Y model of noncompact ty p e and without Euclidean factor. Sublemma 4.26 Y model has r ank one. Pr o of: If r ank ( Y model ) ≥ 2 then the subgroup of translations along a maximal ﬂat in Y model acts on Y as a para b olic subgroup (b ecause no subgroup ∼ = R 2 in I som ( Y ) can con tain axial isometries) and ﬁxes exactly one p oint on ∂ ∞ Y . Maximal ﬂats in Y model con taining parallel singular g eo desics yield the same ﬁxed p oin t in ∂ ∞ Y and it follo ws that H o w ould hav e a ﬁxed p o in t on ∂ ∞ Y , con t r a diction.  It remains to construct the equiv ariant homeomorphism o f b oundaries. Axial isometries in I so m ( Y ) hav e the prop erty that their conjugacy class neve r accum ulates at the iden tity . Therefore if h ∈ H o acts a s a pure parab olic (see deﬁnition 2.10) on Y model then it a cts as a parab olic on Y . Hence the stabilizer of ξ 0 ∈ ∂ ∞ Y model in H o ﬁxes a unique p o in t ξ ∈ ∂ ∞ Y and w e obtain an H o -equiv ariant, and hence con tinuous surjectiv e map ∂ ∞ Y model → ∂ ∞ Y . It m ust b e injectiv e, to o, b ecause any t wo stabilizers of distinct p oin ts in ∂ ∞ Y model generate H o but H o has no ﬁxed po in t on ∂ ∞ Y . This concludes the pro o f of 4.24.  25 Prop osition 4.27 Complete ge o d e s ics in Y don ’t br anch. Pr o of: h ∈ H o acts as a pure parab olic on Y iﬀ it do es so on Y model . The purely parab olic stabilizer N ξ ⊂ H o of ξ ∈ ∂ ∞ Y is a simply connected nilp oten t Lie group and acts simply transitiv ely on ∂ ∞ Y \ { ξ } . Let τ ∈ H o b e any axial isometry acting on Y with attractiv e ﬁxed po in t ξ . Then lim n →∞ τ − n φτ n = e. (24) for all φ ∈ N ξ . Let c b e a geo desic in Y asymptotic to b oth ﬁxed p oin ts of τ a t inﬁnit y and let φ ∈ N ξ b e non-trivial. τ acts as an isometry on the compact cross section of P ( c ) and w e can c ho ose a seq uence n k → ∞ so that d ( c, τ n k c ) → 0. d ( φτ n k c (0) , τ n k c (0)) = d ( τ − n k φτ n k c (0) , c (0)) → 0 implies that φ c is stro ng ly a symptotic to c . These tw o geo desics can’t inte rsect b ecause φ is not elliptic. ( N ξ has no non-t r ivial elliptic elemen ts!) The argument sho ws that distinct stronlgy asymptotic geo desics a r e disjoin t and henc e geo desics in Y don’t branc h.  Pr o of of 1.7: 1.7 is not m uc h more than a reformulation of 4.2. As in the pro of of 3.8 w e deduce from the 2-fold transitivit y of the action of I som ( C ) on ∂ ∞ C that the parab olic stabilizer of an y η ∈ ∂ ∞ C acts transitiv ely on ∂ ∞ C \ { η } . The n C and I som ( C ) satisfy assumption 4 .1 and a ssertion follo ws fro m 4.2 and 4.16.  5 Geo d esically complete Hadamard spaces wit h building b oundary 5.1 Basic prop erties of parallel sets Assumption 5.1 X is a lo c al ly c o mp act Hadama r d sp ac e w ith extendible r ays and ∂ T its X is a spheric al build i n g of dimensio n r − 1 ≥ 1 . Lemma 5.2 Every ﬂat half-plane h in X is c ontaine d in a ﬂat pl a ne. Pr o of: Let c b e the b o undary geo desic of the ﬂat half-plane h and denote ξ ± := c ( ±∞ ). Let η ∈ ∂ ∞ h b e so close to ξ + that the a rc η ξ + in ∂ T its X is con tained in a closed c ham b er, and extend the ray η c (0) to a geo desic c ′ . c ′ b ounds a ﬂat half- plane h ′ whic h con tains ξ + in its ideal b oundary . The canonical isometric em b edding C S ( { ξ + , ξ − } ) ֒ → X ξ + sends h to a ray a nd h ′ to a geo desic extending this ra y . This implies that h is con tained in a ﬂat plane.  Corollary 5.3 F or any ﬂat f ⊂ X the cr oss se ction C S ( f ) is a g ain a lo c al ly c om- p act Hadamar d sp ac e w ith e x tendible r ays, an d ∂ T its C S ( f ) is a spheric a l building of dimension dim ( ∂ T its X ) − dim( f ) . 26 Pr o of: 5.2 implies that for an y geo desic l the cross section C S ( l ) ha s exte ndible ra ys. No w w e pro ceed b y induction on the dimension of f using C S X ( f ) ∼ = C S C S X ( f ′ ) ( C S f ( f ′ )) for ﬂats f ′ ⊂ f .  Corollary 5.4 Ev e ry ﬂat is c ontaine d i n a r -ﬂat. Prop osition 5.5 Supp ose the ge o desic s c 1 , c 2 ⊂ X have a r ay ρ in c ommon . Then ther e ar e two m aximal ﬂats whose interse ction is a half a p artment. Pr o of: Denote ξ := ρ ( ∞ ) = c i ( ∞ ) and ξ i := c i ( −∞ ). There exist geo desics γ i of length π in ∂ T its X joining ξ and ξ i so that their intersec tion γ 1 ∩ γ 2 is a non-degenerate arc ξ η . The geo desics c i pro ject to geo desics ¯ c i in the space of strong asymptote classes X η , and for an y ρ (0)-a n tip o de ˆ η of η t he geo desics ¯ c i are in fact con tained in the pro jection to X η of the cross section C S ( { η , ˆ η } ). The geo desics ¯ c i share a ra y but do not coincide b ecause they ha ve diﬀeren t ideal endp o ints ¯ c i ( −∞ ) = → η ξ i ∈ Σ η ∂ T its X ∼ = ∂ ∞ C S ( { η , ˆ η } ). W e ma y pro ceed b y induction on the dimension of the Tits b oundary of the cross section until w e ﬁnd a ﬂat f so that C S ( f ) has discrete Tits b oundary and con tains tw o g eo desics whose in tersection is a ra y . These geo desics corresp ond to maximal ﬂats in P ( f ) with the desired prop ert y .  Reform ulation 5.6 I f ther e ar e br an ching ge o desics in X then ther e exists a ﬂat f ⊂ X so that ∂ T its f is a wal l in ∂ T its X an d C S ( f ) c ontains br anchin g ge o desics. 5.2 Boundary isomorphisms Deﬁnition 5.7 L et X ′ b e another sp a c e satisfying 5.1. A b oundary isomorphism is a c o ne top olo gy h ome om orphism φ : ∂ ∞ X − → ∂ ∞ X ′ (25) which at the same time is a Tits isometry, i.e. it is an is omorphism of top olo g ic al spheric a l buildings, cf. [BS87]. We denote by I so ( ∂ ∞ X , ∂ ∞ X ′ ) the sp ac e of al l b ound- ary isomorphisms ∂ ∞ X → ∂ ∞ X ′ e quipp e d wi th the c omp act-op en top olo gy, an d by Aut ( ∂ ∞ X ) the top ol o gic al gr oup I so ( ∂ ∞ X , ∂ ∞ X ) . A b o undary isomorphism (25) induces for all simplices τ ⊂ ∂ T its X an isomorphism of top ological buildings Σ τ ∂ T its X − → Σ φτ ∂ T its X ′ . (26) The induced homeomorphisms I so (Σ τ ∂ T its X , Σ ˜ τ ∂ T its X ) → I so (Σ φτ ∂ T its X , Σ φ ˜ τ ∂ T its X ) carry H ol X ( τ , ˜ τ ) to H ol X ′ ( φτ , φ ˜ τ ) and thereby induce isomorphisms of top ological groups H ol X ( τ ) − → H o l X ′ ( φτ ) . (27) 27 Assumption 5.8 In addition to 5 .1 the building ∂ T its X is thick and irr e ducible. According to 5.3, C τ has extendible rays . (Extendibilit y o f ra ys is inherited b y subsets with full ideal b oundary .) If τ ⊂ ∂ T its X is a pa nel then by 3.8 the action of H ol ( τ ) on C τ b y isometries satisﬁes 4.1 and therefore 4.2 applies. In the case that Σ τ ∂ T its X ∼ = ∂ ∞ C τ is homeomorphic t o a sphere, it can b e identiﬁe d with the b oundary of a rank-one symmetric space canonically up to conformal diﬀeomorphism, and Σ φτ X ′ as w ell. In this situation the “diﬀeren tials” (2 6) are confo rmal diﬀeomor- phisms b ecause they a re equiv ariant with resp ect to (27). In the second case that Σ τ ∂ T its X ∼ = ∂ ∞ C τ is disconnected, C τ and C φτ are metric trees and (26) is conformal in the sense that it is induced b y a homothety (4.20). The ideal b oundary ∂ ∞ X , equipp ed with t he cone top ology and Tits metric, is a compact top olo g ical spherical building. The cone t op ology can b e induced b y a metric and this allows us to apply the results fr om [BS87] on a utomorphism groups of top ological spherical buildings. In particular, [BS87, theorem 2.1] implies: Theorem 5.9 (Burns-Spatzier) Aut ( ∂ ∞ X ) is lo c al ly c omp act. W e denote b y F the space of c ham b ers in ∂ T its X . The cone top ology induces a top ology on F whic h makes F a compact space. Lemma 5.10 The r e exist ﬁni tely many chamb ers σ 1 , . . . , σ s such that the map Aut ( ∂ ∞ X ) − → F s \ D iag ; φ 7→ ( φ σ 1 , . . . , φσ s ) (28) is pr op e r 14 . Pr o of: Cho ose σ r +1 , . . . , σ s as the cham b ers of an apartmen t a , and let τ 1 , . . . , τ r b e the panels of σ r +1 . An automo r phism φ is determined b y its eﬀect on a and the spaces Σ τ i ∂ T its X , b ecause ∂ T its X is the con v ex hu ll of the apartmen t a and all c ham b ers adjacen t t o its c hamber σ r +1 15 . Cho ose for each panel τ i a c hamber σ i 6⊂ a with σ i ∩ σ r +1 = τ i . Clearly (28) is con tin uous. Let ( φ n ) be a seq uence in Aut ( ∂ ∞ X ) whose image under (28) is b ounded, i.e. do es not accum ulate at D iag . W e ha v e to sho w that ( φ n ) is b ounded, resp ectiv ely it suﬃces to show that there is a bo unded subseque nce. After passing to a subsequenc e, w e ma y assume that φ n σ i → ¯ σ i with pairwise diﬀerent limits ¯ σ i . Denote ¯ τ i := lim φ n τ i . F or each i ≤ r the sequenc e of confo rmal homeomorphisms Σ τ i ∂ T its X → Σ φ n τ i ∂ T its X con v erges o n a triple of p oin ts (namely on Σ τ i ( a ∪ σ i )) t o an injectiv e limit map and hence sub conv erges uniformly to a (conformal) homeomorphism Σ τ i ∂ T its X → Σ ¯ τ i ∂ T its X . It fo llows that ( φ n ) subconv erges uniformly to a building automorphism.  Consequence 5.11 T he se quenc e ( φ n ) ⊂ Aut ( ∂ ∞ X ) is unb ounde d iﬀ ther e exist adjac e n t chamb ers σ , σ ′ such that φ n σ and φ n σ ′ c onver ge in F to the sa m e cha m b er. 14 D iag denotes the gener alize d dia gonal co nsisting of tupels with at least tw o equal entries. 15 Pro of: The convex hull is a subbuilding B ′ of maxima l dimens io n [KL9 6, prop. 3 .1 0.3]. Since any panel is pro jectively equiv alent to a panel τ i , B ′ is a neighbor ho od of int ( τ ) for any panel τ ⊂ B ′ . W e ca n connect an in terior p oint of an y c ha m b er to a p oint in a b y a geo desic av oiding simplices o f c o dimension ≥ 2. It follo ws that all cham b ers a re con tained in B ′ and B ′ = ∂ T its X . 28 5.3 The case of no branc hing A ma jor part o f the argumen ts in this section follo ws the lines of Gromov ’s pro of of his Rigidity Theorem [BGS85] a nd the study o f top ological spherical buildings in [BS87]. Assumption 5.12 X is a lo c al ly c om p act Hadamar d sp ac e with extendible r ays and ∂ T its X is a thick irr e d ucible spheric al building of dimension r − 1 ≥ 1 . Mor e over we assume in this se ction that c omplete ge o desics in X do not br anc h . F or ev ery p oint x ∈ X there is an in volution ι x : ∂ ∞ X − → ∂ ∞ X whic h maps ξ ∈ ∂ ∞ X t o the other b oundary p oin t of the unique geo desic extending the ra y xξ . Lemma 5.13 ι x ∈ Au t ( ∂ ∞ X ) . Pr o of: The absence of branching implies that ι x is contin uous. By 5.6, each ray emanating from x is con tained in a maximal ﬂat F . ι x restricts o n t he unit sphere ∂ T its F to the antipo dal inv olutio n. Hence ι x maps ev ery c ham b er isometrically to a cham b er and is therefore 1-Lipsc hitz con tin uous with respect to the Tits distance. The claim follow s b ecause ι − 1 x = ι x .  5.13 sho ws that the group Aut ( ∂ ∞ X ) is large. Our aim is to unmask it as the isometry group of a symmetric space. D enote b y I nv the subgroup consisting of all pro ducts of an ev en num b er of in v olutions ι x i . Lemma 5.14 I nv is p ath c onne cte d and it is c ontaine d in every o p en sub gr oup of Aut ( ∂ ∞ X ) . Pr o of: The map X × X → Aut ( ∂ ∞ X ); ( x 1 , x 2 ) 7→ ι x 1 ι x 2 is contin uous and hence I nv is path connected. The second assertion follow s in view of ( ι x 1 ι x 2 )( ι x 2 ι x ′ 2 ) = ι x 1 ι x ′ 2 .  Lemma 5.15 F or any two chamb ers σ 1 , σ 2 in a thick spheric al building B ther e is a c ommo n antip o dal chamb er. R eﬁ n ement: F or any two simplic es of the same typ e 16 ther e is a c ommon antip o d a l simp l e x . Pr o of: Let ˆ σ b e a cham b er an tip o dal t o σ 1 and γ : [0 , π ] → B a unit sp eed geo desic a v oiding co dimens io n- 2 faces with γ (0) ∈ int ( σ 2 ) a nd whic h in tersects int ( ˆ σ ). I f γ ( π ) ∈ ˆ σ then w e are done. Otherwise let τ ⊂ ∂ ˆ σ b e the panel where γ exits ˆ σ . Since B is thic k, there exists a c hamber ˆ σ ′ opp osite to σ 1 so that ˆ σ ′ ∩ ˆ σ = τ . Let γ ′ : [0 , π ] → B b e a unit speed geo desic with γ ′ (0) = γ (0), ˙ γ ′ (0) = ˙ γ (0), whic h agrees with γ up to ˆ σ and then turns through τ in to the in terior of ˆ σ ′ . W e rep eat this pro cedure un til it terminates after ﬁnitely steps and yields a c hamber opp osite to σ 1 and σ 2 . The reﬁneme nt follo ws directly .  16 The typ e of a s implex is its imag e under the canonical (accordeon) pro jection to the model W eyl cham b er ∆ model . 29 Consequence 5.16 F or any simplex τ , I nv acts tr ansitively on the c om p act sp ac e F τ of simplic es of same typ e as τ . In p articular, I nv acts tr an s i tive l y on the c omp act sp ac e F of Weyl cham b ers 17 . No w we in ves tiga te the dynamics on ∂ ∞ X of elemen ts whic h corres p ond to trans- lations (transv ections) along geo desics in symmetric spaces. Lemma 5.17 Supp ose that ρ : [0 , ∞ ) → X i s a r ay asymptotic to ξ and that U ⊂ ∂ T its X is a c omp act set of ξ - a ntip o des. Then ι ρ ( t ) U → { ξ } as t → ∞ . Pr o of: In every Σ η ∂ T its X , η ∈ U , w e c ho ose a n a partmen t α η so that the apartmen ts ∂ ∞ per sp ηξ α η ⊆ Σ ξ ∂ T its X coincide. Conside r sequences t n → ∞ and ( η n ) ⊂ U . W e ha v e to sho w that ι ρ ( t n ) η n → ξ . Let F n b e a maximal ﬂat containing the ray ρ ( t n ) η n and satisfying Σ η n ∂ ∞ F n = α η n . Sublemma 5.18 T he fam ily of ﬂats F n is b ounde d. Pr o of: Assume the con tra ry and, after passing to a subsequence, that η n → η ∈ U . Denote by a the unique apart ment in ∂ T its X con taining ξ , η and so tha t Σ η a = α η . Let R > 0 b e large. F n dep ends con tinuously on t n (b y “no branc hing”), and by decreasing the t n w e can acheiv e that d ( F n , ρ (0)) = R f o r almost all n . Still t n → ∞ if R is c hosen suﬃcien tly large; namely d ( ρ ( t ) η , ρ (0)) is bounded b ecause there exists a geo desic asymptotic to ξ and η . The F n sub con v erge to a maximal ﬂat F with d ( F , ρ (0)) = R and ∂ ∞ F = a . This can’t b e p ossible for a rbitrarily large R b ecause the family of ﬂats with ideal b oundary a is compact, a contradiction.  All ﬂats a rising as limits of ( F n ) are asymp to tic to ξ , η and the an tip o des ι ρ ( t n ) η n of η n in ∂ ∞ F n con v erge to an an tip o de of η , i.e. they con v erge to ξ .  Consequence 5.19 L et c : R → X b e a ge o desi c , ξ ± := c ( ±∞ ) and a t := ι c ( t ) ι c ( − t ) . Then lim t →∞ a t η = ξ + iﬀ ∠ T its ( η , ξ − ) = π . The c onver genc e is uniform on c omp act sets of ξ − -antip o des. Pr o of: By 5.17, ι c ( − t ) η → ξ − uniformly . Then for lar g e t , ι c ( − t ) η and ξ + are antipo des. Applying 5.17 again yields the claim.  Denote by B ( ξ + , ξ − ) ⊂ ∂ T its X the subbuilding deﬁned a s the union of all min- imizing geo desic s with endp oin ts ξ ± , or equiv alently , the union of all apartments con taining ξ ± . There is a folding map (building morphism, see [KL 96, sec. 3.10]) f old : ∂ T its X → B ( ξ + , ξ − ) whic h is uniquely determined b y the pro p ert y that ∠ T its ( f ol dη , ξ − ) = ∠ T its ( η , ξ − ) and − → ξ − ( f ol dη )= − → ξ − η for all η ∈ ∂ T its X with ∠ T its ( η , ξ − ) < π and f ol dη = ξ + if ∠ T its ( η , ξ − ) = π . Reﬁnemen t 5.20 lim t →∞ a t = f ol d . 17 F is the analo g of F¨ urs tenber g b oundary in the symmetric space case. 30 Pr o of: By 5.19 and b ecause a ll a t ﬁx the Tits neighborho o d B ( ξ + , ξ − ) of ξ − p oin twise .  Prop osition 5.21 Aut ( ∂ ∞ X ) is a semisimp le Lie gr oup whose identity c omp onent has trivial c enter. Pr o of: 1. Aut ( ∂ ∞ X ) is a Lie gr oup: Let G ′ ⊆ Aut ( ∂ ∞ X ) b e an op en subgroup, c a geo desic, ξ ± = c ( ±∞ ) and U + a neigh b orho o d of ξ + whic h is c hosen so small that all p oints in U + with the same ∆ model -direction (t yp e) as ξ + are ξ − -an tip o des (using the low er semicon tin uity of Tits distance). Suppo se H ⊂ G ′ is an inv ariant subgroup con tained in the neigh b orho o d { φ ∈ G ′ : φξ + ∈ U + } of e . Then H ξ + consists of ξ − -an tip o des. Hence H ξ + = a t H a − 1 t ξ + = a t H ξ + → { ξ + } as t → ∞ , th us H ξ + = ξ + . Since F ix ( H ) is G ′ -in v a rian t and conv ex with resp ect to the Tits metric it fo llows from 5.16 that F ix ( H ) = ∂ ∞ X and H = { e } . So there a re neighborho o ds of the iden tit y in G ′ whic h don’t con tain non-trivial in v arian t subgroups. 2.2 0 implies tha t Aut ( ∂ ∞ X ) is a Lie group. Sublemma 5.22 Eve ry non-trivial is o metry φ of a thick spheric al building B diﬀer- ent fr om a spher e c arries so me p oint to an antip o d e. Pr o of: W e ma y a ssume without loss of generality that B has no spherical join factor. If the assertion w ere not true then φ would b e homotopic to the identit y and therefore preserv e ev ery apartment and hence ev ery simplex, so φ = id .  2. Sem isimplicity: Supp ose that A is an in v ar ia n t ab elian subgroup of Aut o ( ∂ ∞ X ). Let a ∈ A b e a non- t rivial elemen t and c ho ose a simplex τ − suc h tha t τ − and aτ − are opp osite (using 5.22). τ − then has in v olutio n- in v ar ia n t 18 t yp e. Let c : R → X b e a geo desic with c ( −∞ ) ∈ int ( τ − ) and τ + the simplex con taining c (+ ∞ ). Set a n := ι c ( n ) ι c ( − n ) ∈ I nv and b n := a n aa − n ∈ A . 5.19 implies lim n →∞ b n τ = τ + for all simplices in the op en subset W = { τ ∈ F τ − : τ and τ + are opp osite } of F τ − . In view of 5.15, W a nd the attractor τ + are uniquely determined by the dynamics of ( b n ) and therefore are preserv ed b y the cen tralizer of ( b n ) in Aut o ( ∂ ∞ X ). Th us A has ﬁxed p oin ts on F τ − . 5.1 6 implies that t he action of A on F τ − is trivial. The ﬁxed p oint set of A on ∂ T its X includes the con v ex hull of all simplices in F τ − and this is the whole building ∂ T its X by irreducibilit y 19 . So A = { e } . This sho ws that all ab elian in v arian t subgroups of Aut o ( ∂ ∞ X ) are trivial, hence also the solv able in v ariant subgroups. This ﬁnishes the pro o f of 5.21.  As a conseque nce of t he prop osition, there is a symmetric space X model of non- compact t yp e and an isomorphism Aut o ( ∂ ∞ X ) ∼ = − → I som o ( X model ) (29) of Lie groups. 18 The type of a s implex is involution-invariant if its an tip o dal simplices ha ve the same type , or equiv a le nt ly , if the type is ﬁxed b y the self-isometr y of ∆ model which is induced b y the inv olution of the spherical Coxeter complex. 19 The convex hull of simplices of the same in volution-inv ariant type in the spher ical Coxeter complex is a subsphere, hence everything by irreducibility . 31 Lemma 5.23 The c entr alize r of every involutive b oundary automorphis m ι x is c om - p act. Pr o of: Supp ose that ( φ n ) is an unbounded sequence in the cen tra lizer of ι x . Then there are adjacen t c ham b ers σ , σ ′ so that lim φ n σ = lim φ n σ ′ (b y 5.11). The sequence of conformal diﬀeomorphisms (diﬀeren tials) Σ σ ∩ σ ′ ∂ T its X → Σ φ n ( σ ∩ σ ′ ) ∂ T its X is un- b ounded and con v erges ev erywhere except in at most one p oint to a constan t map Σ σ ∩ σ ′ ∂ T its X → Σ lim φ n ( σ ∩ σ ′ ) ∂ T its X . Denote b y s ⊂ ∂ T its X the w all spanned b y the opp osite panels σ ∩ σ ′ and ι x ( σ ∩ σ ′ ). It follo ws that for all half-apartments h ⊂ ∂ T its X with ∂ h = s with the excep tio n of at most one half-apartmen t h 0 , the limits lim φ n | h exist a nd hav e the same half apartmen t ¯ h a s image. Since ∂ T its X is thick , w e ﬁnd an ι x -in v a rian t apartmen t a containing s but not h 0 . So φ n | a con v erges to a non-injectiv e map a → ¯ h comm uting with ι x , i.e. sending an tip o des to antipo des. Such a map can’t exist and w e reac h a con t r adiction.  Sublemma 5.24 L et X 0 b e an irr e ducible symmetric sp ac e. Eve ry a utomorp h ism of I som o ( X 0 ) is the c onjugation by an isometry, i.e. I som ( X 0 ) ∼ = Aut ( I som o ( X 0 )) . Pr o of: X 0 = G/K .  The inv olution ι x ∈ Aut ( ∂ ∞ X ) induces b y conjugation an inv olutiv e automor- phism o f Aut o ( ∂ ∞ X ), hence an in volutiv e isomorphism of I som o ( X model ) via (29), and as a consequence of 5.23, the corresp onding in v olutive isometry of X model is the reﬂection at a p oin t Φ( x ) ∈ X model . W e obta in a prop er con tinuous map Φ : X − → X model . (30 ) Another direct consequence is that pro ducts ι x ι x ′ of tw o in v olutions correspond to translations (or the identit y) in I som ( X model ). F or an y ﬂat F ⊂ X whose ideal b oundary ∂ ∞ F is a singular sphere w e denote by T F ⊂ Aut o ( ∂ ∞ X ) the subset of all ι x ι x ′ with x, x ′ ∈ F . Lemma 5.25 As a subset of I som ( X model ) , T F is the gr oup of tr anslation s along a ﬂat F Φ of the same dimension as F . Mor e ov e r r ank ( X model ) = r . Pr o of: Let a ⊂ ∂ T its X b e an apartment con taining ∂ ∞ F , σ a cham b er in a and ξ 1 , . . . , ξ r the v ertices of σ . Moreov er denote by τ i the panel o f σ opp o site to ξ i , and b y ˆ σ , ˆ τ i , ξ i the resp ectiv e an t ip o dal ob jects in a . An automorphism φ o f ∂ T its X whic h ﬁxes a p oin twis e is determined b y its actions on the spaces Σ τ i ∂ T its X . W e therefore obtain an em b edding S tab Aut ( ∂ ∞ X ) ( a ) ֒ → r Y i =1 H omeo (Σ τ i ∂ T its X ) . An automorphism whic h ﬁxes the subbuilding ∂ T its P ( { ξ i , ˆ ξ i } ) is determined b y it s action on Σ τ i ∂ T its X alone and we g et an em b edding S tab Aut ( ∂ ∞ X ) ( ∂ T its P ( { ξ i , ˆ ξ i } )) ֒ → H omeo (Σ τ i ∂ T its X ) . 32 Eac h Σ τ i ∂ T its X is iden tiﬁed with b o undary of a rank-one symmetric space. φ ∈ S tab Aut ( ∂ ∞ X ) ( a ) acts on Σ τ i ∂ T its X b y a conformal diﬀeomorphism (compare the dis- cussion in section 5.2) which ﬁxes at least the t wo p oint set Σ τ i a . This diﬀeomorphism is hence con tained in a subgroup of the conformal group isomorphic to R × cpt . As a consequence, S tab Aut ( ∂ ∞ X ) ( a ) top o logically em b eds in to a group ∼ = R r × cpt and the subgroups H i = S tab Aut ( ∂ ∞ X ) ( ∂ T its P ( { ξ i , ˆ ξ i } )) embed in to R × cpt . Moreov er H i cen tralises H j for i 6 = j . It follo ws that all translations in I som ( X model ), wh ich corre- sp ond to pro ducts ι x ι x ′ suc h t ha t x, x ′ lie on a geo desic asymptotic to ξ i and ˆ ξ i , lie in the same 1 -parameter subgroup T i . Moreov er the T i comm ute with eac h other. Since x 7→ ι x is prop er, the ﬁrst assertion fo llows. If F is a maximal ﬂat with ∂ ∞ F = a then the centralizer of T F is con tained in S tab Aut ( ∂ ∞ X ) ( a ) and thus contains no subgroup ∼ = R r +1 . Hence r ank ( X model ) can’t b e greater than r .  Consequen tly , (30) sends maximal ﬂats to maximal ﬂats. F lats whose ideal b ound- aries are singular spheres arise as in tersections of maximal ﬂats and hence g o to sin- gular ﬂats. It follows from irreducibilit y that Φ restricts to a homothety on ev ery ﬂat and clearly the scale factors for restrictions to diﬀeren t ﬂats agree. Since X is geo desically complete by assumption, ev ery pair of p oints lies in a maximal ﬂat (5.4) and it follo ws that Φ is a homothet y . This conclude s the pro of of the main result of this section: Theorem 5.26 L et X b e a lo c al ly c omp act Hadamar d sp ac e with extendible g e o desics and who se Tits b oundary is a thick irr e ducible s p heric al building of dime n sion r − 1 ≥ 1 . If c omplete ge o des i c s in X don ’t br anch then X i s a Riemannian symmetric sp ac e of r ank r . The argument ab o v e also sho ws that , for an irreducible symmetric space X 0 of rank ≥ 2, the Lie groups I som ( X 0 ) and Aut ( ∂ ∞ X 0 ) hav e equal dimens ion and hence the natural embedding I som ( X 0 ) ֒ → Aut ( ∂ ∞ X 0 ) is op en and induces an isomorphism of iden tit y components. Of course, more is true: Theorem 5.27 (Tits) L et X 0 b e an irr e ducible symmetric sp ac e of r ank ≥ 2 . Then the natur al emb e d ding I som ( X 0 ) − → Aut ( ∂ ∞ X 0 ) ( 31) is an isomorph ism. Pr o of: Let ψ b e an automorphism of ∂ ∞ X 0 . W e ha v e to sho w tha t ψ is induced b y an isometry of X 0 . ψ induces an auto mo r phism α of Aut o ( ∂ ∞ X 0 ) ∼ = I som o ( X 0 ) whic h sends the stabilizer of an apartment a to the stabilizer of ψ a , i.e. it sends the group of translations along the ﬂat F a ﬁlling in the apartmen t a ( ∂ ∞ F a = a ) to the translations along F ψa . The isometry Ψ inducing α (5.24) th us satisﬁes Ψ F a = F ψa , i.e. ∂ ∞ Ψ( a ) = ψ a for all apartmen ts a and it follow s ∂ ∞ Ψ = ψ .  5.27 implies 1.3 in the smo oth case. 33 5.4 The case of branc hing Assumption 5.28 X is a lo c al ly c om p act Hadamar d sp ac e with extendible r ays and ∂ T its X is a thick irr e d ucible spheric al building of dimension r − 1 ≥ 1 . Mor e over we assume in this se ction that s o m e c omplete ge o desics br anch in X . Note that no w w e can’t exp ect a big group Aut ( ∂ ∞ X ) of b oundary automor- phisms. There exist completely asymmetric Euclidean buildings of rank 2. Our ap- proac h is based on the observ ation that nev ertheless the cross sections of all parallel sets are highly symmetric (3.8). 5.4.1 Disconnectivit y of F ¨ ursten b erg b oundary The aim of this section is: Prop osition 5.29 If fo r some p anel σ o f B = ∂ T its X the sp ac e Σ σ B is total ly d is- c onne cte d , then this is true for al l p anels . Pr o of: W e ﬁrst consider the case when B is one-dimensional. l denotes the length of a W eyl arc and irreducibilit y implies π /l ≥ 3. The vertices (singular p oints) of B can b e tw o-colo ured, sa y blue and red, so that adja cent v ertices ha ve diﬀeren t colo urs. The distance of t wo v ertices is an ev en multiple of l iﬀ they hav e t he same colour. According to 3.8, the Hadamard spaces C ξ satisfy 4.1 for a ll v ertices ξ ∈ B . 4.2 tells that Σ ξ B is homeomorphic to a sphere of dimension ≥ 1, a Can tor s et or a ﬁnite set with at least 3 elemen ts (b ecause B is t hic k). V ertices ξ 1 , ξ 2 ∈ B of the same colour are pro jectiv ely equiv alen t 20 and therefore the spaces of directions Σ ξ i B are homeomorphic. If π /l is o dd t hen an y tw o an tip o dal v ertices ha v e diﬀeren t colours and the Σ ξ B a re homeomorphic for all v ertices ξ . If π /l is ev en (and hence ≥ 4 b y irreducibilit y), w e hav e t o rule out the po ssibilit y that Σ ξ B is disconnected for blue v ertices ξ and connected for red v ertices. Let us assume that this w ere the case. Sublemma 5.30 I f Σ ξ B is a spher e for r e d vertic e s ξ then Σ η B c a n ’t b e ﬁnite for blue vertic es η . Pr o of: Assume that Σ ξ B is a sphere for red v ertices ξ and Σ η B is ﬁnite fo r blue v ertices η . 1. R e d vertic e s ξ , ξ ′ of distanc e 4l lie i n the same p ath c omp onent of the singular set S ing ( B ) : There exists a red v ertex η with d ( ξ , η ) = d ( ξ ′ , η ) = 2 l . ξ , ξ ′ , η lie in an apartmen t a (b ecause 4 l ≤ π ). Let ˆ η b e the antipo de of η in a . Since Σ η B is path-connected w e can contin uously deform the geo desic η ξ ˆ η to the g eo desic η ξ ′ ˆ η , so ξ and ξ ′ can b e connected b y a red path. 2. F or every r e d vertex ξ 0 the (r e d) distanc e sp h er e S 2 l ( ξ 0 ) is p ath-c onne cte d: Let ξ 1 , ξ 2 b e red vertice s with d ( ξ i , ξ 0 ) = 2 l . There is a ve rtex ξ ′ 2 in the same path comp onen t of S 2 l ( ξ 0 ) as ξ 2 suc h that d ( ξ 1 , ξ ′ 2 ) = 4 l . (Deform as in 1. using an an tip o de of ξ 0 .) 20 F or 1-dimens io nal s pherical buildings pr oje ctive e quivalenc e is the equiv alence relation for ver- tices generated by antipo dality . 34 3. S 2 l ( ξ 0 ) is a manifold of the sa me dimension a s Σ ξ 0 B : W e in tro duce lo cal co- ordinates on S 2 l ( ξ 0 ) near ξ as follows. Let η b e the midpo int of ξ 0 ξ , i.e. d ( ξ 0 , η ) = d ( η , ξ ) = l . Cho ose an tip o des ˆ ξ 0 of ξ 0 and ˆ η of η . F or ξ ′ ∈ S 2 l ( ξ 0 ) near ξ the midp oin t η ′ of ξ 0 ξ ′ is close to η and d ( η ′ , ˆ η ) = π , d ( η ′ , ˆ ξ 0 ) = d ( ξ ′ , ˆ η ) = π − l . → ˆ ξ 0 η ′ and → ˆ η ξ ′ are con tin uous lo cal co ordinates for ξ ′ and it follows that S 2 l ( ξ 0 ) is a manifo ld of the same dimension as Σ ξ 0 B . 4. Since Σ ξ 0 B em b eds in to S 2 l ( ξ 0 ) it follo ws that S 2 l ( ξ 0 ) ∼ = Σ ξ 0 B via the map ξ 7→ → ξ 0 ξ , and S 2 l ( ξ 0 ) is contained in the susp ension B ( ξ 0 , ˆ ξ 0 ). This implies that the cardinalit y o f Σ η B is 2 and contradicts thic kness.  F or the rest of the pro of of 5.29 we assume that Σ η B is a Cantor set fo r blue v ertices η and Σ ξ B is a sphere for red ξ . Sublemma 5.31 L et ξ , η , η ′ ∈ B b e distinct vertic es (of the same c olor) with d ( ξ , η ) = d ( ξ , η ′ ) = π − 2 l and let U b e a neighb orho o d of ξ . Then ther e exists a v e rtex ξ ′ ∈ U satisfying d ( ξ ′ , η ) = π − 2 l and d ( ξ ′ , η ′ ) = π . (32) Pr o of: Let ζ b e the v ertex with ξ η ∩ ξ η ′ = ξ ζ and ω the vertex on ζ η adjacent to ζ . Extend ω ξ b ey ond ξ to a geo desic ω ˆ ω of length π . Σ ω B ha s no isolated p oin ts a nd w e can pick a geo desic γ connecting ω and ˆ ω so tha t the initial v ector Σ ω γ is close to → ω ζ and the vertex ξ ′ ∈ γ with d ( ξ ′ , ω ) = d ( ξ , ω ) lies in U . By construction, (32) holds.  Sublemma 5.32 L et γ : ( − ǫ, ǫ ) → S ing ( B ) b e a c ontinuous p ath in the r e d singular set a n d ξ b e a r e d vertex so that d ( ξ , γ (0)) = π − 2 l . Then d ( ξ , γ ( t )) = π − 2 l for t close to 0 . Pr o of: Let η b e a blue v ertex adja cen t to ξ so that d ( η , γ (0)) = π − l . The set of v ertices at distance π − l from η is op en in the singular set and so d ( η , γ ( t )) = π − l for t close to 0 . Since Σ η B is totally disconnecte d we ha v e − → η γ ( t )= → η ξ for small t , hence the claim holds.  Sublemma 5.33 T he p ath γ is c onstant. Pr o of: It suﬃces to sho w that γ is lo cally constan t. There are neighbor ho o ds U o f ξ and V of η := γ (0) so that d ( ξ ′ , η ′ ) ≥ π − 2 l for all vertice s ξ ′ ∈ U and η ′ ∈ V (b ecause the Tits distance is upp er semicon tin uous). W e assume without loss of generalit y that γ do es not leav e V . Then 5.32 implies that d ( ξ , γ ( · )) ≡ π − 2 l . If γ w ere not lo cally constan t w e could choose t so that η ′ := γ ( t ) 6 = η . Applying 5.31 there exists ξ ′ ∈ U so that ( 3 2) holds. But 5.32 implies also that d ( ξ ′ , γ ( · )) ≡ π − 2 l . Hence d ( ξ ′ , η ′ ) = π − 2 l , con tradicting (32). Thus γ is lo cally constan t.  Hence, the set of red ve rtices has trivial path comp onen ts. But since π /l > 2, the space of directions Σ ζ B for any v ertex ζ con tinuously embeds in to the blue singular set as w ell as into the red singular set. Therefore Σ ζ B can’t b e connected for any 35 v ertex ζ , con tra diction. Hence Σ ζ B m ust b e a Can tor set for all vertice s ζ . This concludes the pro o f of 5.29 in the 1-dimensional case. Without muc h transpiration o ne can deduce the assertion in the general case dim ( B ) ≥ 1: Let σ, τ b e panels of t he same cham b er with angle ∠ ( σ, τ ) < π / 2. Then the 1-dimensional top ological spherical building Σ σ ∩ τ B is irreducible a nd w e ha ve canonical homeomorphisms: Σ σ B ∼ = Σ Σ σ ∩ τ σ Σ σ ∩ τ B , Σ τ B ∼ = Σ Σ σ ∩ τ τ Σ σ ∩ τ B , Moreo v er, Σ σ ∩ τ B is the ideal b oundary of a Hadamard space satisfying 5.28 with r = 2, namely of the cross section C S ( f ) for an y ( r − 2)-ﬂat f with ∂ ∞ f ⊃ σ ∩ τ . Therefore w e can apply our assertion in the 1-dimensional case and see that Σ σ B is a Can tor set if and only if Σ τ B is. Since B is irreducible, for a n y t w o panels σ, σ ′ exists a ﬁnite sequence of panels σ 0 = σ, σ 1 , . . . , σ m = σ ′ so that a n y tw o successiv e σ i are a djacen t with angle less than π / 2 21 . This ﬁnishes the pro of of 5.29.  5.4.2 The str ucture of parallel sets Consider a ( r − 1)- ﬂat w ⊂ X whose b oundary at inﬁnit y is a w all in the spherical building ∂ T its X . F or an y panel τ ⊂ ∂ ∞ w , C τ is canonically isometric to the con v ex core of C S ( w ). By 4.2 and 4 .16, the follo wing three statemen ts are equiv alent: • C S ( w ) is the pro duct of a metric tree times a compact Hadamard space. • ∂ ∞ C S ( w ) is homeomorphic to a Can tor set. • Some geo desics branc h in C S ( w ) 22 . By 5.28 and 5.6, there exists a ( r − 1)- ﬂat w so tha t C S ( w ) contains branc hing geo desics. ∂ ∞ w is a w all in ∂ T its X and for an y panel σ ⊂ ∂ ∞ w we ha v e that Σ σ ∂ T its X ∼ = ∂ ∞ C S ( w ) is a Can tor set. 5.29 implie s that Σ σ ∂ T its X is a Can t o r set for all panels σ in ∂ T its X and hence: Lemma 5.34 ∂ ∞ C S ( w ) is tr ee × compact for al l ( r − 1) -ﬂats w . 5.4.3 Pro of of the main result 1.2 Theorem 5.35 L et X b e a lo c al ly c omp ac t Hadamar d sp ac e with extend i b le r ays and whose Tits b oundary is a thick irr e ducible spheric al buildin g of di m ension r − 1 ≥ 1 . If ther e ar e br an ching c omplete ge o d e sics in X then X splits as the pr o duct of a Euclide an building of r ank r times a c omp act Hadam a r d sp ac e. Pr o of: W e ﬁrst inv estigate the lo cal structure of X . F o r ev ery p oint x ∈ X w e hav e the canonical 1-Lipsc hitz con tinuous pro jection θ x : ∂ T its X → Σ x X 21 Otherwise w e could sub div ide the panels of the model W eyl cham b er in to tw o families so that panels in diﬀerent families are orthogonal; this w o uld imply reducibility . 22 This equiv alent to branching of geo des ic s in C τ . 36 whic h assigns to ξ ∈ ∂ T its X the direction → xξ ∈ Σ x X . Therefore, if ξ , ˆ ξ ∈ ∂ T its X are x -an tip o des, i.e. ∠ x ( ξ , ˆ ξ ) = π , then θ x restricts t o an isometry on eve ry geo desic in ∂ T its X of length π connecting ξ and ˆ ξ . By o ur assumption of extendible ra ys, ev ery ξ has x -an tip o des, and it f o llo ws that θ x restricts t o an isometry on ev ery simplex . If σ , ˆ σ are op en c ham b ers in ∂ T its X whic h a r e x -o pp osite in the sense that there exist x -an tip o des ξ ∈ σ and ˆ ξ ∈ ˆ σ , then θ x restricts to an isometry o n the unique apartmen t in ∂ T its X con taining σ, ˆ σ and w e call its image an ap artment in Σ x X . If σ 1 , σ 2 are op en simplices wh ose θ x -images inte rsect then there exists a simplex ˆ σ whic h is x -opp osite to b oth σ i . It follo ws that the θ x -images of the sphere s span ( σ i , ˆ σ ) and therefore the θ x σ i coincide. Hence the θ x -images of op en simplices in ∂ T its X a r e disjoin t or they coincide and w e call them simplic es or fac es in Σ x X . Sublemma 5.36 T he θ x -images of adjac ent chamb ers σ 1 , σ 2 ⊂ ∂ T its X ar e c on tain e d in an ap artment. (They may c oincid e .) Pr o of: Let ξ b e a p oin t in the op en pa nel σ 1 ∩ σ 2 and ˆ ξ an x -antipo de. θ x is isometric on the half-apartments h i = span ( ˆ ξ , σ i ) b ecause it is isometric on σ i and ∠ x ( ξ , ˆ ξ ) = π . The union H i of the ra ys with initial p oin t x and ideal endp oint ∈ h i is a half- r -ﬂat in X . The ( r − 1)-ﬂats ∂ H i coincide and our assumption on cross sections of parallel sets allo ws tw o p ossibilities: Either H 1 ∪ H 2 is a r -ﬂat and the θ x σ i are adjacen t c ham b ers in an apartmen t. Or the H 1 ∩ H 2 is a non-degenerate ﬂat strip and the θ x σ i coincide.  As a consequence, the centers of adjacen t c ha mbers in Σ x X are uniformly sepa- rated and the compactness of Σ x X implies that the n umber of simplices in Σ x X is ﬁnite. Sublemma 5.37 Any two simplic e s in Σ x X ar e c ontaine d in an ap artment. Pr o of: Since in ∂ T its X any tw o simplices are con ta ined in a n apartment it suﬃces to pro v e the follow ing statemen t: ( ∗ ) If c 1 is a chamb er c ontaine d in an ap a rtmen t a and c 2 is a chamb e r s o that c 2 ∩ a is a p anel, then ther e exists an ap artment a ′ c ontainin g the chamb ers c 1 and c 2 . The rest then follo ws b y induction. T o prov e ( ∗ ) we consider the hemisphere h ⊂ a with c 1 ⊂ h and c 2 ∩ a ⊂ ∂ h . Applying 5.36 to the c ha m b er c 2 and the adjacen t cham b er in h w e see that there is a geo desic γ : [0 , π ] → Σ x X con tained in im ( θ x ) wh ich starts in int ( c 2 ), pass es through c 2 ∩ a in to h and sta ys in h for the r est of the time and inters ects int ( c 1 ) on its wa y . The regular endp o in ts o f γ span a unique apartmen t a ′ in Σ x X , and a ′ ⊃ c 1 ∪ c 2 .  As a consequenc e, im ( θ x ) is a conv ex, compact subse t of Σ x X and hence itself a CA T(1)- space. The spherical building structure on ∂ T its X induces a spherical building structure on im ( θ x ). Consequence 5.38 F or any two p oints ξ 1 , ξ 2 ∈ ∂ ∞ X the angle ∠ x ( ξ 1 , ξ 2 ) c an take only ﬁnitely many values whi c h dep end on the typ es θ ∂ T its X ξ i ∈ ∆ model . Let us denote b y S u n x the union of all ra ys emanating fr o m x . Lemma 5.39 Any two sets S un x and S un y ar e disjoint or c oincide. 37 Pr o of: Assume that x 6 = y a nd y ∈ S un x . W e pic k an ideal point ξ ∈ ∂ ∞ X and sho w that the ray y ξ is con tained in S un x : First w e extend y x t o a ra y y xη . Then w e c ho ose a minimal g eo desic connecting → y x and → y ξ inside im ( θ y ) and exte nd it b ey ond → y ξ to a geo desic α of length π . Denote the endp oint by u . There is a cham b er σ in Σ y X whic h contains the end of α near u , and w e lift σ to a c hamber ˜ σ in ∂ T its X . θ y restricts to an isometry on any apartmen t ˜ a ⊂ ∂ T its X whic h con tains ˜ σ and η . Therefore ˜ a b ounds a ﬂat F whic h contains x and a ray y ξ ′ with → y ξ ′ = → y ξ in Σ y X . The ra ys y ξ ′ and y ξ initially coincide (by 5.3 8) and therefore S un x ∩ y ξ is half-op en in y ξ tow ards ξ . Since it is clearly closed, it follows that S un x con tains the ray y ξ and hence S un y ⊆ S un x . Now the segmen t xy is contained in a geo desic. I.e. x ∈ S un y and a naloguously S un x ⊆ S u n y . This show s tha t y ∈ S u n x iﬀ S un x = S un y . It follo ws that if z ∈ S un x ∩ S un y then S un x = S un z = S un y , hence the claim.  It follow s that the subsets S un x are minimal closed con v ex with full ideal b oundary ∂ ∞ S un x = ∂ ∞ X . Consequen tly they are parallel and, b y the second part of 2.2 , X decomp oses as a pro duct of Z × compact . Z is a geo desically complete Hadamard space a nd it remains to v erify that it carries a Euclidean building structure. Its Tits b oundary ∂ T its Z = ∂ T its X and the spaces of directions Σ z Z carry spherical building structures mo delled on the same Co xeter complex ( S, W ) so that the maps θ z : ∂ T its Z → Σ z Z are building morphisms, i.e. they are compatible with the direction maps to the mo del W eyl c hamber ∆ model . θ ∂ T its Z = θ Σ z Z ◦ θ z . (33) (The buildings Σ z Z are in general not thick.) Cho ose a Euclidean r -space E , iden tif y ∂ T its E ∼ = S and let W af f ⊂ I som ( E ) b e the full inv erse image o f W under the canonical surjection r ot : I som ( E ) → I som ( S ). Up to isometries in W af f w e can pic k a canonical c hart E → F f o r ev ery ma ximal ﬂat F ⊂ Z . The co o r dinate c hanges will b e induced b y W af f . Since geo desic segmen ts are extendible they are contained in maximal ﬂats and in view of (33) we can assign to them w ell-deﬁned ∆ model - directions. The directions clearly satisfy the angle rigidity prop ert y (cf. section 2.3.2) and w e hence ha v e a Euclidean building structure on Z mo delled on the Euclidean Co xeter complex ( E , W af f ). (If one wishes, one can reduce the a ﬃne W eyl gr o up and obtain a canonical thic k Euclidean building structure.) This concludes the pro of of 5.35.  Pr o of of 1.2: Put in 5.26 and 5.35. Stir gen tly .  5.5 Inducing b oundary isomorphisms b y homotheties: Proof of 1.3 Pr o of of 1.3: By 1.2, X and X ′ are symm etric spaces or Euclidean buildings. The F ¨ urstenberg b o undar y of X is a Can tor set iﬀ X is a Euclidean building. Hence X, X ′ are either bot h symmetric or b o th buildings. The assertion in the symmetric case is the con ten t of 5.27. W e may therefore a ssume that X and X ′ are thick irreducible Euclide an buildings of rank r ≥ 2. Then for an y ﬂat f the cross section C S ( f ) is a Euclidean building 38 of rank r − dim( f ), and has no Euclidean factor if f is singular. F or all geo desics l the canonical em b eddings C S ( l ) ֒ → X l ( ±∞ ) of cross sections into spaces of strong asymptote classes are now surjectiv e isometries, and fo r ev ery ξ ∈ ∂ ∞ X , C ξ ∼ = X ξ is a Euclidean building of rank r − 1 whic h splitts o ﬀ a Euclidean de Rha m factor of dimension dim( τ ξ ) where τ ξ denotes the simplex containing ξ as interior p oin t. In particular, for all panels τ ⊂ ∂ T its X , C τ is a metric tree. As explained in section 5.2, the diﬀeren tials (26 ) of (1) are b oundary maps of homotheties C τ − → C φτ (34) and these comm ute with the system of natural p ersp ectivity iden tiﬁcations (16). The assertion o f 1.3 follows if w e can pin dow n ev ery v ertex of X b y data at inﬁnity . This is ac heiv ed by the follo wing b o wtie construction suggested b y Bruce Kleiner: A b owtie ⊲ ⊳ consists of a pair of opp osite c ham b ers σ ⊲ ⊳ and ˆ σ ⊲ ⊳ , of v ertices y i ∈ C τ i for each panel τ i ⊂ σ ⊲ ⊳ and v ertices ˆ y i ∈ C ˆ τ i for the opp osite panels ˆ τ i so tha t persp τ i ˆ τ i y i = ˆ y i holds. ⊲ ⊳ determines a v ertex in X as follows: σ ⊲ ⊳ and ˆ σ ⊲ ⊳ are con tained in the ideal b oundary of a unique maximal ﬂat F ⊲ ⊳ ⊂ X and ev ery pair y i , ˆ y i determines a w all w i ⊂ F ⊲ ⊳ . The r w alls w i in tersect in a unique v ertex x ⊲ ⊳ . W e sa y lo osely that ⊲ ⊳ is c ontaine d in the ﬂat F ⊲ ⊳ . W e call tw o b ow ties ⊲ ⊳ and ⊲ ⊳ ′ pre-adjacen t if σ ⊲ ⊳ ∩ σ ′ ⊲ ⊳ = τ r , ˆ σ ⊲ ⊳ = ˆ σ ′ ⊲ ⊳ and ˆ y i = ˆ y ′ i for all i . (Then also y r = y ′ r holds.) There is an ob vious inv olutio n on the space of b o wties and a n equally ob vious action of the p erm uta tion group S r and w e call tw o b ow ties adjac ent if they are pre-adjacen t mo dulo thes e op erations. Adjacen t b ow ties determine the same v ertex. Adjacency spans an equiv alence relatio n on the set of b o wties whic h w e denote b y “ ∼ ”. Lemma 5.40 ⊲ ⊳ ∼ ⊲ ⊳ ′ iﬀ x ⊲ ⊳ = x ⊲ ⊳ ′ . Pr o of: Clearly ⊲ ⊳ ∼ ⊲ ⊳ ′ implies x ⊲ ⊳ = x ⊲ ⊳ ′ . T o prov e the con v erse, let us assume that x ⊲ ⊳ = x ⊲ ⊳ ′ . W e start with a special case: Sublemma 5.41 I f ⊲ ⊳ and ⊲ ⊳ ′ lie in the same a p artment then ⊲ ⊳ ∼ ⊲ ⊳ ′ . Pr o of: It is enough to c hec k the case when σ ⊲ ⊳ and σ ⊲ ⊳ ′ share a panel, i.e. without loss of generalit y τ 1 = τ ′ 1 , ˆ τ 1 = ˆ τ ′ 1 , y 1 = y ′ 1 and ˆ y 1 = ˆ y ′ 1 . Since y 1 , ˆ y 1 are v ertices there exists a half- r -ﬂat H ⊂ X so that H ∩ F = ∂ H = w 1 . If ⊲ ⊳ ′′ is adjacen t to ⊲ ⊳ and ⊲ ⊳ ′′′ is adjacen t to ⊲ ⊳ ′ so t hat σ ⊲ ⊳ ′′ = σ ⊲ ⊳ ′′′ ⊂ ∂ ∞ H then ⊲ ⊳ ′′ and ⊲ ⊳ ′′′ are adjacen t. So ⊲ ⊳ ∼ ⊲ ⊳ ′ .  Sublemma 5.42 L et ˆ σ b e a chamb er in ∂ T its X . The n ther e exists a b owtie ⊲ ⊳ ′′ ∼ ⊲ ⊳ so that ˆ σ ⊲ ⊳ ′′ = ˆ σ . Pr o of: It is enough to treat the case when ˆ σ is adjacen t to ˆ σ ⊲ ⊳ in ∂ T its X . After replacing ⊲ ⊳ by an equiv a len t b o wtie (e.g. contained in the same maximal ﬂat) w e ma y assume that θ x ⊲ ⊳ ˆ σ and θ x ⊲ ⊳ σ ⊲ ⊳ are opp osite cham b ers in Σ x ⊲ ⊳ X . Then w e can c ho ose ⊲ ⊳ ′′ adjacen t to ⊲ ⊳ .  W e reﬁne the previous sublemma: 39 Sublemma 5.43 L et F b e a maxim al ﬂat an d ⊲ ⊳ a b owtie with σ ⊲ ⊳ ⊂ ∂ ∞ F and x ⊲ ⊳ ∈ F . L et ˆ σ b e any chamb er in ∂ T its X . Then ther e exists another b owtie ⊲ ⊳ ′′ ∼ ⊲ ⊳ so that ˆ σ ⊲ ⊳ ′′ = ˆ σ and σ ⊲ ⊳ ′′ ⊂ ∂ ∞ F . Pr o of: Again, w e may a ssume without loss of generalit y that the cham b er ˆ σ is adjacen t to ˆ σ ⊲ ⊳ . If θ x ⊲ ⊳ ˆ σ is opp o site to the c ham b er θ x ⊲ ⊳ σ ⊲ ⊳ in Σ x ⊲ ⊳ X then w e can c ho ose ⊲ ⊳ ′′ adjacen t to ⊲ ⊳ . Otherwise let σ ⊂ ∂ ∞ F b e the c hamber adjacen t to σ ⊲ ⊳ so that θ x ⊲ ⊳ σ is opp osite to θ x ⊲ ⊳ ˆ σ and denote b y ⊲ ⊳ ′′ the b o wtie with σ ⊲ ⊳ ′′ = σ , ˆ σ ⊲ ⊳ ′′ = ˆ σ and x ⊲ ⊳ ′′ = x ⊲ ⊳ . Then ⊲ ⊳ ′′ is equiv alent to a b ow tie contained in F ⊲ ⊳ and hence to ⊲ ⊳ .  T o ﬁnish the pro of of 5.40 we can ﬁrst replace ⊲ ⊳ ′ b y an equiv alen t b owtie so that σ ⊲ ⊳ = σ ⊲ ⊳ ′ (5.42) and then replace it in a second step so that ⊲ ⊳ ′ and ⊲ ⊳ lie in the same apartmen t (5.4 3 ). Hence ⊲ ⊳ ′ and ⊲ ⊳ are equiv alen t (5.41).  It follows that equiv alence classes of b owties in X corresp o nd to v ertices. Since (1) induces a map b etw een the spaces of b ow ties in X and X ′ whic h preserv es the equiv a lence relatio n “ ∼ ”, it thereb y induces a map Φ : V er t ( X ) → V er t ( X ′ ) on v ertices. Φ maps all ve rtices in a singular ﬂat f ⊂ X to the v ertices of a singular ﬂat f Φ ⊂ X ′ so that φ ( ∂ ∞ f ) = ∂ ∞ f Φ . Since X and X ′ are irreducible buildings, Φ extends to a homothety Φ : X → X ′ and ∂ ∞ Φ = φ . This concludes the pro of of 1.3.  5.6 Extension of Mosto w and Prasad Rigidit y to singular spaces of nonp ositiv e c u rv ature: Pro of of 1.5 Pr o of of 1.5: W e argue as Mosto w [Mos73]. A Γ- p erio dic ﬂat is a maximal ﬂat whose stabilizer in Γ acts cocompactly . Due to results of Bo rel and Ballmann-Brin, Γ-p erio dic ﬂats lie dense in the space of all ﬂats in X model . By our a ssumption, there is a Γ-equiv a rian t con tin uous ma p Φ : X model − → X . It is a quasi-isometry and carries Γ-p erio dic ﬂa ts in X model to Γ- p erio dic quasi-ﬂats in X with uniform quasi-isometry constan ts. If a quasi-ﬂat is Hausdorﬀ close to a ﬂat then it lies in a tubular neigh b orho o d of this ﬂat whose radius is uniformly b ounded in terms of t he quasi-isometry constan t s ([Mos73, Lemma 13.2] for sym metric spaces and [KL96] f or buildings). Density and unifo r mity imply tha t Φ maps ev ery ﬂat in X model uniformly close to a ﬂa t in X and with this information one can construct a Γ-equiv ariant boundary isomorphism Φ ∞ : ∂ ∞ X model − → ∂ ∞ X . By 1.2 X is a symmetric space or Euclidean building, and by 1.3, after suitably rescaling the irreduc ible factors of X model , Φ ∞ is induced b y a Γ-equiv aria n t isometry X model → X .  40 References [Al57] A.D. Aleksandro v, ¨ Ub er eine V er al lge m einerung der Riemannschen Ge ome- trie, Sc hriften des F o r sch ungsinstituts f. Mathematik 1 (1957), 33-84 . [Ba85] W. Ballmann, Nonp ositively c urve d m anifolds of higher r ank, Ann. Math. 122 (1985), 597- 6 09. [Ba95] W. Ballmann, L e ctur es on sp ac es of nonp osi tive curvatur e , DMV-Seminar notes, v ol. 2 5 , Birkh¨ auser 1995. [BGS85] W. Ballmann, M. G romo v, V. Sc hro eder, Man i f o lds of Nonp ositive Curva- tur e, Birkh¨ auser 1985. [BS87] K. Burns and R. Spatzier, On top olo gic al Tits buildings and their classi ﬁ c a- tion, Publ. IHES 65 (1987), 35-59. [EO73] P . Eb erlein and B. O ’Neill, Visibility man i f o lds, P aciﬁc J. Math. 4 6 (197 3), 45-109. [Eb88] P . Eb erlein, Symmetry Diﬀe omo rphism Gr oup of a Manifold of Nonp ositive Curvatur e II, Indiana U. Math. J. 37, no. 4 (1988), 735 -752 . [Gl52] A. Gleason, Gr oups without sma l l sub gr oups, Ann. Math. 56 (1952), 193-2 12. [Gr87] M. G romo v, Hy p erb olic Gr oups, in: Essay s in Group Theory , MSRI Publica- tions v ol. 8, Springer 1987. [Gr93] M. Gromo v, Asymptotic invarian ts for inﬁnite gr oups, in: Geometric group theory , London Math. So c. lecture note series 182 , 1993. [Ka] F. I. Karp elevi ˇ c, The ge ometry of ge o d e sics and the eig e nfunctions of the Beltr ami- L aplac e op er ator o n symmetric sp ac es, Amer. Math. So c. T ransla- tions 14 (1985), 51-19 9. [KL96] B. Kleiner and B. Leeb, Rigidity of quasi- i s o metries for symmetric sp ac es and Euclide an buildings, pr eprint, Bonn a pril 1, 1996, to app ear in Publications IHES. [Le95] B. Leeb, 3-manifold s with(out) metrics of n o np ositive curvatur e, In v entiones math. 122 (1995), 277- 289. [Ma91] G. Margulis, Discr ete sub gr oups of semisimple Lie gr oups, Springer 1991 . [MZ55] D . Mon tgomery and L. Zippin, T op olo gic a l tr an sformation gr oups, In ter- science Publishers, New Y ork 1955. [Mos73] G. D. Mostow, Str ong rigidity of lo c al ly symme tric sp ac es, Princeton UP 1973. [Pra79] G. Prasad, L a ttic es i n semisimple gr oups ove r lo c al ﬁelds, Studies in algebra and n um b er theory , Adv. Math. Suppl. Studies 6 (197 9 ). 41 [Ron89] M. R onan, L e ctur es on B uildings, P ersp ectiv es in Mathematics v ol. 7, Aca- demic Press 1989. [Ti74] J. Tits, Buildings of Spheric a l T yp e an d Finite BN-Pairs, LNM 38 6, Springer 1974 (2nd ed. 1986). [Y a53] H. Y amab e, A gene r alization of a the or em of Gle ason, Ann. Math. 58 (1953), 351-365 . 42

A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment