Coarse Equivalences of Euclidean Buildings

Coarse Equiv alences of Euclid ean Building s Lin us Kramer and Ric hard M . W eiss ∗ With an app endix b y Jeroen Sc hillew aert and Ko en Struyv e Abstract W e prov e the following r igidity results. Coarse eq uiv alences betw een metr ically co mplete Eu- clidean buildings preserve spherical buildings a t inﬁnity . If all irreducible factors hav e dimension at least t wo, then coa rsely equiv alen t Euclidean buildings are isometric (up to scaling fa ctors); if in addition none of the irreducible factors is a Euclidea n co ne, then the is ometry is unique and ha s ﬁnite dista nce f rom the coars e equiv alence. The app endix sho ws ho w these r esults ca n b e extended to non-complete Euclidea n buildings. W e pro v e co arse (i.e. quasi-isometric) rigidit y results for l eaﬂess trees (simplicial trees and R -trees with extensible geo desics) and, more generall y , for d iscrete and nondiscrete Euclidean buildings. F or trees, a k ey ingredient is a certain equiv ariance condition. Our main results are as follo ws. Theorem I L et G b e a gr o up acting isometric a l ly on two metric al ly c omp lete le a ﬂess tr e es T 1 , T 2 . Assume that ther e is a c o arse e qui valenc e f : T 1 ✲ T 2 , that T 1 has at le ast 3 ends and that the induc e d map ∂ f : ∂ T 1 ✲ ∂ T 2 b etwe en the ends of the tr e es is G -e quivariant. If the G -action on ∂ T 1 is 2 -tr ansitive, then (after r esc aling the metric on T 2 ) ther e is a G -e qu i variant isometry ¯ f : T 1 ✲ T 2 with ∂ f = ∂ ¯ f . If T 1 has at le ast two br anch p oints, then ¯ f is uniqu e and has ﬁnite distanc e fr om f . The precise result is 2.18 b elo w , where w e also consider pr o ducts of trees and Euclidean spaces. Without the e quiv ariance condition, quasi-isometric rigidit y fails. Th e letters X and H written in ﬁnitely large are, f or example, trees whic h are coarsely equiv alen t without b eing isometric. Note, too, that an y t wo simplicial trees of ﬁ nite constan t v alence are coarsely equiv alen t without b eing necessarily isometric [34]. Our next results are concerned w ith (p ossibly n ondiscrete) Euclidean buildings. Theorem I I L et X 1 and X 2 b e Euclide an buildings whose spheric al buildings at inﬁnity ∂ cpl X 1 and ∂ cpl X 2 ar e thick. Supp ose that f : X 1 × R m 1 ✲ X 2 × R m 2 is a c o arse e qui valenc e. Then m 1 = m 2 and ther e is a c ombinatorial isomor phism f ∗ : ∂ cpl X 1 ✲ ∂ cpl X 2 b etwe en the spheric al buildings at inﬁnity which is char acterize d by the fact that for an aﬃne ap artment A ⊆ X 1 the f -image of A × R m 1 has ﬁnite Hausdorﬀ distanc e fr om f ∗ ( A ) × R m 2 . This is 5.16 b elo w. W e remark that the b oundary map f ∗ is constru cted in a combinato rial w a y from f . In general, a coarse equiv alence b et w een CA T(0)-spaces w ill not induce a map b etw een the resp ectiv e Tits b oundaries. Theorem I I I L et f : X 1 × R m ✲ X 2 × R m b e as in The or em II and assume in addition that X 1 has no tr e e factors and that X 1 and X 2 ar e metric al ly c omplete. Then ther e is (p ossibly after r esc aling ∗ Kramer and W eiss were supp orted b y the SFB 878 and DF G Pro ject K R 166 8/7. 1 the metrics on the de R ham factors of X 2 ) an isometr y ¯ f : X 1 ✲ X 2 with ( ¯ f × id R m ) ∗ = f ∗ . Put f ( x × v ) = f 1 ( x × v ) × f 2 ( x × v ) . If none of the de Rha m f actors of X 1 is a E u clide an c one over its b oundary, then ¯ f is uniqu e and d ( f 1 ( x × y ) , ¯ f ( x )) is b ounde d as a function of x ∈ X 1 . F or a more general state ment see 5.21 b elo w. Kleiner a nd Leeb p ro v ed results similar to Theorems I I and I I I in [25, 1.1.3] a nd [31, 1.3] under the additional h yp othesis that the Euclidean buildin gs are simplici al and lo cally ﬁnite or their buildings at inﬁnity satisfy the Moufang condition. Their wo rk extended Mosto w-Prasad rigidit y [36]. Our results oﬀer in particular an alternativ e approac h to these imp ortan t ac hiev emen ts. Outline of the pap er In t he ﬁrst s ection we collect so me basic facts ab out metric sp aces, nonp ositiv e curv ature, ultrapro d ucts and ultralimits. The sec ond section is concerned w ith trees. W e deriv e in particular the structure of leaﬂess trees that admit an isometry group that is 2-transitiv e on the ends. Th eorem I is then an application of this structur al result, com bined w ith Morse’s Lemma. In the third section we collect th e facts ab out spherical buildin gs and p ro jectivitie s that we need. The fourth section is devote d to Euclidean buildin gs and their lo cal and global stru cture. In the ﬁfth section we com bin e all previous results. W e ﬁr st p ro v e Theorem I I, using the ‘higher dimens ional Morse Lemma’. Then w e com bine Th eorem I and Theorem I I in order to p ro v e Theorem I I I. T he pro of relies, among other things, on Tits’ rigidit y result [46, Thm. 2] of ‘ecologica l’ (tree-preserving) b ound ary isomorphisms of Euclidean b uildings. 1 Some metric geometry W e recall a few notions and facts ab out metric spaces that w ill b e needed later. Let ( X, d ) b e a metric space. F or r > 0 and x ∈ X w e p ut B r ( x ) = { y ∈ X | d ( x, y ) < r } and ¯ B r ( x ) = { y ∈ X | d ( x, y ) ≤ r } . F or a su bset Y ⊆ X w e put B r ( Y ) = S { B r ( y ) | y ∈ Y } . W e call Y ⊆ X b ounde d if Y i s con tained in some suﬃcien tly large ball, Y ⊆ B r ( x ) for some x ∈ X , and w e call Y c ob ounde d i f X = B r ( Y ) for some r ≥ 0. Let f : X ✲ Y b e a map b etw een metric sp aces. If ther e is a constan t r > 0 suc h th at d ( f ( u ) , f ( v )) ≤ r d ( u, v ) holds for all u, v ∈ X , we call f an r -Lipschitz map . A map f whic h preserv es distances is called an isometric emb e dding ; if f is in addition ont o, it is called an isometry . The group of all isometries of X onto itself is the isometry gr oup Isom( X ). An isometric action of a grou p G on X is a homomorphism G ✲ Isom( X ). W e call su c h an action (or group) b ounde d if s ome (or, equiv alently , eve ry) G -orbit is b ounded in X . T he action is called c ob ounde d if there is a cob ounded orbit. 1.1 CA T( κ ) spaces A ge o desic in a metric space X is an isomet ric emb edding γ : J ✲ X , where J ⊆ R is a closed in terv al. The image γ ( J ) is then cal led a ge o desic se gment . If the domain of γ is J = R , then γ ( R ) is also called a ge o desic line , and if J = [0 , ∞ ), then γ ([0 , ∞ )) is called a ge o desic r ay. If any t wo p oin ts of X are con tained in some geo desic segment , X is called a ge o desic sp ac e . If ev er y geo desic γ : J ✲ X admits a geo desic extension ¯ γ : R ✲ X , w e sa y that X has extensib le ge o desics . 2 A geo desic space X is called a CA T(0) sp ac e if n o geod esic tr iangle in X is thic k er than its comparison triangle in Euclidean space R 2 , see [7, I I .1]. More generally , a geodesic s pace is called a CA T( κ ) sp ace, for κ ∈ R , if no geodesic triangle in X is th ic ker than its comparison triangle in the complete simp ly connected Riemannian 2-manifold M κ of constan t sectional curv atur e κ . F or κ > 0, the space M κ is a sphere and the condition is only required for geo desic tr iangles of perim eter less than 2 π √ κ (and the existence of geo desics is only r equired b et ween p oin ts at distance less than π √ κ ). If ( X, d ) is CA T( κ ), then the same s pace is also CA T( κ ′ ) for all κ ′ ≥ κ [7, I I.1.12]. If r > 0 and if ( X, d ) is C A T( κ ), then the rescale d space ( X , r d ) is CA T( κ/ √ r ). If K is a metrically complete con ve x subset in a CA T(0) space X , then there is a 1-Lipsc h itz retraction π K : X ✲ K whic h maps x ∈ X to the unique closest p oin t in K [7, I I.2.4]. The m etric completion of a CA T( κ ) space is again a CA T( κ ) sp ace. One imp ortan t fact ab out complete CA T(0) spaces is the Br uhat-Tits Fixed P oint Theorem: ev ery boun ded isometry group in suc h a sp ace has a ﬁxed p oin t [7, I I.2.8]. 1.2 Controlled maps W e no w recall some notions from coarse geometry [37]. A map f : X ✲ Y b et w een metric spaces is called c ontr ol le d if there is a monotonic real function ρ : R ≥ 0 ✲ R ≥ 0 suc h that d Y ( f ( x ) , f ( y )) ≤ ρ ( d X ( x, y )) holds for all x, y ∈ X . If in addition the pr eimage of every b oun ded s et is b ounded, then f is called a c o arse map . Neither f nor ρ is requ ired to b e con tinuous. Note that the image of a b ounded set un der a con trolled map is b ounded. Tw o maps g , f : X ✲ ✲ Y b et w een metric spaces ha v e ﬁnite dist anc e if the set { d Y ( f ( x ) , g ( x )) | x ∈ X } is b oun ded. T his is an equiv alence relation which lea ds to the c o arse metric c ate gory whose ob jects are metric spaces and wh ose morph isms are equiv alence classes of coarse maps. A c o arse e quivalenc e is an isomorphism in th is category . 1.3 Lemma If f : X ✲ Y a nd g : Y ✲ X are con trolled and if g ◦ f has b ound ed d istance from the iden tit y map on X , then f is coarse. In particular, f is a coarse equiv alence if f ◦ g also has b ound ed distance from the identit y . Pr o of. Su pp ose th at f ( Z ) is b ounded . Since g is control led, g ( f ( Z )) is also b oun ded, g ( f ( Z )) ⊆ B r ( x ) for some x ∈ X . No w Z is co nta ined in some s -neigh b orho o d of g ( f ( Z )), so Z ⊆ B r + s ( x ) is b ound ed. ✷ 1.4 Qua si-isometries A cont rolled map f with co ntrol function ρ ( t ) = t i s the sa me as a 1 -Lipsc hitz map. If f admits a con trol function ρ ( t ) = ct + d, with c ≥ 1 and d ≥ 0 , then f is call ed lar ge-sc ale Lipschitz . A coarse equiv alence is called a q u asi-isometry if b oth the map and its coarse inv erse are large-scale Lips c h itz. More generally , we call f a quasi-isometric emb e dding if f is a qu asi-isometry b etw een X and f ( X ) ⊆ Y . A r ough isometry is a coarse equiv alence w here the con tr ol functions in b oth directions are of the form ρ ( t ) = t + d (suc h maps are sometimes called c o arse isometries ). A map wh ic h has ﬁn ite d istance from an isometry is an example of a r ough isometry . 3 In the class of geo desic sp aces, controlle d maps are essent ially th e same as large-scale Lipschitz m aps. 1.5 Lemma Let f : X ✲ Y be con trolled. If X is geod esic, th en f is large-scale Lip s c h itz. In particular, ev ery coarse equiv alence b et wee n geo desic spaces is automatical ly a quasi-isometry . Pr o of. F or x, y ∈ X let d ( x, y ) = m + s , with m ∈ N and 0 ≤ s < 1. Let γ b e a geo desic from x to y . Then d ( f ( γ ( j )) , f ( γ ( j + 1 ))) ≤ ρ (1) for j = 0 , . . . , m − 1, so d ( f ( γ (0)) , f ( γ ( m + s ))) ≤ mρ (1) + ρ ( s ) ≤ ( m + s ) ρ (1) + ρ (1). ✷ 1.6 Hausdorﬀ distance Two (nonempty) subs ets U, V of a metric space X ha ve Hausdorﬀ distanc e at most r if U ⊆ B r ( V ) and V ⊆ B r ( U ) . In this case we write H d ( U, V ) < r . W e deﬁn e for U, V , W ⊆ X the Hausd orﬀ distance [19 , VI I I § 6] as H d ( U, V ) = inf { r > 0 | H d ( U, V ) < r } and H d W ( U, V ) = H d ( U ∩ W, V ∩ W ) . F or examp le, a nonempt y su bset is b ounded if and only if it h as ﬁnite Hausdorﬀ distance from some p oint . More generally , w e sa y that V domina tes U if U ⊆ B r ( V ) for some r > 0, and we write then U ⊆ H d V . This deﬁ nes a p reorder on the subs ets of X . If f : X ✲ Y is co ntrol led with con trol function ρ , and if U ⊆ B r ( V ), then f ( U ) ⊆ B ρ ( r ) ( V ). S o if f is a coarse equiv alence, then U and V hav e ﬁ nite Hausdorﬀ distance if and only if f ( U ) and f ( V ) h a ve ﬁn ite Hausd orﬀ d istance, and U dominates V if and only if f ( U ) d ominates f ( V ). No w we turn to ultralimits and asymptotic cones of metric s paces. These constructions generalize p oint ed Gromo v-Hausdorﬀ conv ergence of metric sp aces. The adv ant age of ultralimits is that they exist eve n if Gromo v-Hausdorﬀ conv ergence f ails, and that these spaces alwa ys hav e go o d metric and functorial prop erties. In cont rast to [24, Sec. 3.], [25, 2.4], [7, I .5], [37, 7.5], [18, 3.29], we follo w the original approac h of v an den Dries and Wilkie [47], whic h is b ased on elemen tary model theory and ultrapro du cts. The adv ant age of this is viewp oint is that it kee ps tr ac k of the global s tructure of spaces, and th at it allo ws us in a natural wa y to include some fu rther structur e, such as metric balls, apartmen ts, geodesics, or Hausd orﬀ distance. W e r emark that similar tec hniques ha v e b een used in Banac h space theory for the last 50 ye ars. In order to mak e the pap er self-con tained, w e brieﬂy review some elemen tary f acts ab out lan- guages, stru ctures, and ultrapro ducts. They can b e foun d in any ‘old-fashioned’ textbo ok on mathe- matical logic, such as [2] or [12]. 1.7 Languages and structures Supp ose that L is a ﬁr st-order language. S uc h a language is a set consisting of sym b ols for ﬁn itary functions, relations and co nstant s. The language is allo w ed to b e inﬁnite. A sp eciﬁc example whic h will b e i mp ortan t here is the language L of = { + , · , ≤ , 0 , 1 } of ordered ﬁelds. An L -structur e is a tuple A = ( A, F, R , C ) consisting of a n onempt y set A (the u ni v erse of the structure), a set F of ﬁnitary functions, a s et R of ﬁnitary relatio ns, and a set C ⊆ A of constants. In addition, w e ha v e an interpr etation which assigns to ev ery function/relation/constan t sym b ol in L a fu nction/relatio n/constan t in A . F or example, the reals form an L of -structure R if w e interpret the sym b ols + , · , 0 , 1 , ≤ in th eir usual wa y . Ab using notation, w e denote symb ols and their interpretatio ns 4 b y the same letters—this should cause no confusion, as the interpretati on will alw ays b e the natural one. Metric spaces ﬁt well into this concept. Th e relev ant language is L ms = L of ∪ { d, X , R } . If ( X, d ) is a metric space, then we consider th e L ms -structure X w ith u niv erse X . ∪ R , fu nctions + , · , d , relations R, X, ≤ , and constan ts 0 , 1. The unary r elation X ( p ) sa ys that p is a p oint in X and the unary relation R ( t ) says that t is a real num b er. Th e bin ary f unction d giv es the distance on p oin ts in X , while the b inary functions + , · giv e the usual arithmetic op erations in th e real num b ers. 1.8 F orm ulas F rom a language L we build f orm ulas by the standard ru les of ﬁrst-order logic, using the logical sym b ols ∃ , ∀ , ˙ = , ¬ , → and an inﬁnite set of symbols for v ariables. V ariables in a formula whic h are not b ound b y a quantiﬁer are called fr e e . W e write ϕ ( x ) for such a form u la with a fr ee v ariable x . Abusing notatio n, x may also denote a ﬁnite tuple of free v ariables (and w e also allo w tacitly that the formula ϕ ( x ) con tains no f ree v ariables at all). If we w ant to interpret a form ula in a giv en L -structur e, we ﬁrs t ha v e to substitute the free v ariables by elements of th e un iv erse, that is, w e plug in sp eciﬁc v alues a for th e free v ariables x . F or this su bstitution, we write ϕ ( a ). A form u la without free v ariables is called a sentenc e . If a sen tence ϕ holds in the L -str ucture A , one writes A | = ϕ . F or example, we ha ve in the metric sp ace X that X | = ∀ u, v , w  X ( u ) ∧ X ( v ) ∧ X ( w ) → d ( x, z ) ≤ d ( x, y ) + d ( y , z ))  that is, the triangle inequ alit y holds for all triples of p oin ts in X . The form ula ϕ ( x 1 , x 2 ) =  X ( x 1 ) ∧ X ( x 2 ) ∧ ( d ( x 1 , x 2 ) ≤ 1)  with fr ee v ariables x 1 , x 2 holds in the metric s pace X if we substitute p oin ts u, v for x 1 and x 2 whose distance is at most 1. 1.9 Ultrapro ducts S upp ose that K is a countably inﬁnite set and th at D is a n onprincipal ultra- ﬁlter on K . This means that D is a collecti on of sub sets of K , co nt aining all coﬁnite subsets of K , whic h is closed u nder ﬁnite inte rsections, closed un der going up (if a ∈ D and K ⊇ b ⊇ a , then b ∈ D ), and wh ic h con tains for eve ry subset a ⊆ K either a or its co mplement K − a , bu t not b oth. Using Zorn’s Lemma, it is easy to see that non principal ultraﬁlters exist [12, 4.1.3]. Su pp ose that ( A k ) k ∈ K is a family of L -structures A k = ( A k , F k , R k , C k ). W e in tro duce an equiv alence relation on K -sequences ( a k ) , ( b k ) ∈ Q k A k b y putting ( a k ) ∼ D ( b k ) if { j ∈ K | a j = b j } ∈ D. The resulting set of e quiv alence classes is the ultr apr o duct A D = Q k A k /D . On the functions, relations and constants in Q k F k , Q k R k and Q k C k w e introd uce the same equiv alence relation. T he resulting L -structure A D =  A D , F D , R D , C D  is th e ultr apr o duct of the family ( A k ) k ∈ K . If all structures in the family are equal, one calls A D also an u ltr ap ower . Ultrapro ducts are useful b ecause of the follo w ing tw o facts. Firstly , Los’ Theorem sa ys that A D has the s ame ﬁ rst-order p rop erties as ‘the ma jorit y of the A k ’. The second f act is that the ultrapro d uct is ‘h uge’ and conta ins ‘str ange’ element s: it is ω 1 -saturated (this is essen tially the same as the ‘o v erspill’ men tioned in [47, Sec. 3]). F or the next tw o theorems we assu me that ( A k ) k ∈ K is a coun table family of L -structures and that D is a nonp rincipal u ltraﬁlter on K . 5 1.10 The orem ( Los’ Theorem) Let ϕ ( x ) b e an L -form ula (p ossibly with a free v ariable x ). Let a = ( a l ) ∈ Q k A k . Then A D | = ϕ ( a ) if and only if { j ∈ K | A j | = ϕ ( a j ) } ∈ D. (If th ere is no free v ariable in ϕ this means that ϕ holds in A D if and only if the set of indices k for whic h ϕ holds in A k is in D .) Pr o of. The pro of is an easy induction on the complexit y of formulas in [12, 4.1.9] [2, § 5 2.1] [22, 9.5.1]. ✷ The follo win g fact is an eleme nta ry consequence. 1.11 Lemma If A is a ﬁ nite stru cture, then th e ultrap ow er A D is isomorp hic to A . ✷ If w e pu t A k = R = ( R , + , · , ≤ , 0 , 1), w e end u p with the ﬁeld of nonsta ndar d r e als R D = ( ∗ R , + , · , ≤ , 0 , 1) . By Los’ T heorem, this is a real clo sed ﬁeld. If eac h X k is a metric space, then X D is a generalized metric space, where the distance function tak es it v alues in the nonstandard reals ∗ R . W e will come bac k to these g eneralized metric spaces b elo w. V arian ts of the follo wing result are folklore. F or the sak e of completeness, we in clude a pr o of. 1.12 The orem ( ω 1 -saturation of ultrapro ducts) Let Φ b e coun table set of L -formulas in the free v ariable x . S upp ose that f or ev ery ﬁ n ite subset Ψ ⊆ Φ there exists an K -sequen ce ( a k ) su c h that A D | = ψ ( a ) holds for all ψ ( x ) ∈ Ψ . Then there exists an K -sequence ( a k ) su c h that A D | = ϕ ( a ) holds for all ϕ ( x ) ∈ Φ . Pr o of. W e ma y assume that K = N and that Φ is coun tably in ﬁnite. Accordingly , let Φ = { ϕ k ( x ) | k ∈ N } b e an enumeratio n of Φ. W e h a ve b y Los’ Theorem 1.10 that K n = { k ∈ N | k ≥ n and A k | = ∃ x ϕ 0 ( x ) ∧ · · · ∧ ϕ n ( x ) } ∈ D . Since T n ≥ 0 K n = ∅ , there exists for eac h i ∈ K 0 a largest n ( i ) with i ∈ K n ( i ) . F or k ∈ N − K 0 w e c ho ose a k ∈ A k arbitrarily . F or k ∈ K 0 w e choose a k ∈ A k in such a w ay that A k | = ϕ 0 ( a k ) ∧ · · · ∧ ϕ n ( k ) ( a k ) holds. W e claim th at the K -sequence a = ( a k ) h as the r equired prop ert y . Let n ∈ N . F or all k ∈ K n w e ha ve that n ≤ n ( k ), whence A k | = ϕ n ( a k ). Thus A D | = ϕ n ( a ) holds. ✷ As an illustration, consid er the coun table set of L of -form ulas ϕ n ( x ) =  x > 0 + 1 + 1 + · · · + 1 | {z } n summands  . By 1.12 , there are elemen ts in ∗ R wh ic h satisfy th ese formulas sim ultaneously . This means that there are elements in ∗ R which are bigger than every natural num b er. In other w ords, ∗ R is a non- ar chime de an r e al close d ﬁeld . W e call an elemen t r ∈ ∗ R ﬁnite if | r | ≤ n holds for s ome n ∈ N . It is readily v eriﬁed that the ﬁn ite elements form a v aluation ring ∗ R ﬁn ⊆ ∗ R . Its maximal ideal is the set ∗ R inf = { r ∈ ∗ R | | r | ≤ 1 n +1 for all n ∈ N } of inﬁnitesimal elements . One can show that the quotient ﬁeld ∗ R ﬁn / ∗ R inf is order-isomorph ic to R [41, 23.8]. The corresp ond ing ring epim orphism std : ∗ R ﬁn ✲ R 6 assigns to every ﬁ nite nonstandard r eal x ∈ R ﬁn its so-called stand ard part st d ( x ) ∈ R . W e note also that ∗ R con tains canonically a cop y of R (repr esen ted by constant K -sequences of reals). 0 ✲ ∗ R inf ✲ ∗ R ﬁn ✛ ✲ R ✲ 0 1.13 Ultralimits and asymptotic cones F or the remainder of th is section, D w ill d enote a nonprincipal ultraﬁlter on the countably in ﬁnite set K . W e noted ab ov e that the ultrapro d uct X D of a family of metric sp aces is a generalized metric space, whose d istance f unction tak es its v alues in the nonstandard r eals 1 . Let p b e a p oin t in th e ultrapr o duct X D and let r b e a p ositiv e nonstandard r eal. Let X ( p,r ) D =  q ∈ X D | 1 r d ( p, q ) ∈ ∗ R ﬁn  . Then ˜ d = std ◦ 1 r d is a r eal-v alued pseudometric X ( p,r ) D × X ( p,r ) D std ◦ 1 r d ✲ R , where p oin ts with inﬁnitesimal 1 r d -distance ha ve ˜ d -distance 0. Id en tifying p oin ts at ˜ d -distance 0, w e obtain a metric space wh ic h w e denote by C ( X D , p, r ), C ( X D , p, r ) × C ( X D , p, r ) ˜ d ✲ R . This is the ultr alimit of the family ( X k ) k ∈ K with resp ect p and r . If we represen t the n onstandard real r and the p oin t p by K -sequences ( r k ) k ∈ K and ( p k ) k ∈ K , resp ective ly , then these are the sequences of scaling factors and basep oints as in [25, 2.4.2] and [37, 7.5]. In th e notation of th ese authors, we h a ve C ( X D , p, r ) = D - lim 1 r k X k . If r is inﬁnite and all X k are equal to one ﬁxed metric sp ace X , then C ( X D , p, r ) is the asymptotic c one of X . It is imm ediate that ultralimits commute w ith metric p ro duct decomp ositions. F rom the surjection std : ∗ R ﬁn ✲ R w e hav e in particular an isometry C ( ∗ R n , p, r ) ∼ = R n for an y c hoice of p and r . Ultralimits h a v e man y go o d prop erties. F or example, they are spherically complete (and in particular complete, [15, § 32]). Spherical completeness means that ev ery nested sequ ence ¯ B t 0 ( x 0 ) ⊇ ¯ B t 1 ( x 1 ) ⊇ ¯ B t 2 ( x 2 ) ⊇ ¯ B t 3 ( x 3 ) ⊇ · · · of closed balls has nonemp t y intersectio n. 1.14 Lemma The u ltralimit C ( X D , p, s ) of an y coun tably inﬁnite family of metric spaces ( X k ) k ∈ K is s p herically complete. Pr o of. Let ¯ B t 0 ( x 0 ) ⊇ ¯ B t 1 ( x 1 ) ⊇ ¯ B t 2 ( x 2 ) ⊇ ¯ B t 3 ( x 3 ) ⊇ · · · b e a nested sequence of closed balls in C ( X D , p, r ). W e consider the canonical su rjection X D ⊇ X ( p,r ) D ✲ C ( X D , p, r ) 1 Generalized metric spaces, where the metrics tak e v alues in some ordered a b elian group app ear already in H ausdorﬀ [19, VI I I § 5 II I]. See also [14, Ch. 1.2] [33, I I.1]. 7 whic h identiﬁes p oin ts whose 1 r d -distance is inﬁ nitesimal. F or eac h x n w e choose a preimage z n ∈ X ( p,r ) D . F or k, ℓ ∈ N we p ut E k ,ℓ = n z ∈ X D | 1 r d ( z , z k ) ≤ t k + 1 ℓ +1 o . This family has the ﬁnite in tersection prop erty . Indeed, we h a ve z m ∈ E k ,ℓ for all m ≥ k . Adding constan t symb ols z n and t n to our language L ms , w e ma y apply Theorem 1.12. The coun table set of form ulas ϕ k ,ℓ ( x ) =  1 r d ( x, z k ) ≤ t k + 1 ℓ +1  can b e satisﬁed b y a single element z ∈ X D , that is, z ∈ T k ,ℓ E k ,ℓ . The image of th is element in C ( X D , p, r ) is cont ained in T n ≥ 0 ¯ B r n ( x n ). ✷ No w w e supp ose that eac h space X k is CA T( κ k ). Let κ denote the nonstand ard real co rresp onding to the K -sequence ( κ k ). W e also ﬁ x a p ositiv e n onstandard real r . Th e next resu lt is w ell-kno wn [24, 3.6] [25, 2.2.4]; from the viewp oin t of ultrapro ducts and Los’ Theorem, it is almost trivial. 1.15 Lemma F or ev ery real num b er λ with λ ≥ √ r κ , the space C ( X D , p, r ) is CA T ( λ ) . In particular, C ( X D , p, r ) is CA T (0) if eac h X k is C A T (0) . Pr o of. T his is ob vious from Los’ T heorem 1.10 and the fact th at the CA T ( κ ) condition can b e stated in the language L ms . The details are as follo ws. As in the standard deﬁnition of the C A T( κ ) condition [7, I I.1], w e pu t D κ = ∞ for κ ≤ 0 and D κ = π √ κ for κ > 0. F or nonstandard reals s < t we deﬁne the ‘nonstandard inte rv al’ [ s, t ] = { x ∈ ∗ R | s ≤ x ≤ t } , and w e deﬁne ‘nonstandard geo desics’ in X D in the ob vious w a y as isometric injections of s uc h non- standard interv als. It is clear from the fact that the X k are CA T( κ k ) sp aces and from Los’ Theorem that an y t w o p oin ts u, v ∈ X D with d ( u, v ) < D κ can b e joined by a nonstand ard geodesic. W e put ε = 0 if κ = 0 and ε = κ/ | κ | else. Let M ε denote the simply connected complete Riemannian surface of co nstant secti onal curv ature ε , with its metric d ε , and let M ε D denote its u ltrap o w er. F or κ 6 = 0 w e deﬁne a metric d κ on M ε D b y d κ = 1 √ | κ | d ε . Again, w e see fr om Los’ Theorem 1.10 that triangles of p erimeter less th an 2 D κ in ( X D , d ) are not thick er than triangles in ( M ε D , d κ ). Thus X D is a ‘non- standard CA T( κ ) s pace’: it s atisﬁes the usual CA T( κ ) comparison triangle condition, except that the mo del space M ε is replaced by its nonstandard version M ε D . If we rescale the metric on X D b y the factor 1 r , we obtain a nonstandard CA T( √ rκ ) space. If we iden tify in this space ( X D , 1 r d ) p oint s at inﬁnitesimal distance, nonstandard geod esics b ecome ordinary geod esics. The CA T( λ ) inequ alit y also remains v alid for all real λ ≥ √ r κ . Thus C ( X D , p, r ) is a CA T( λ ) sp ace. ✷ Asymptotic cones ma y b e used to ‘make large-scale Lipsc h itz maps con tin uous’ as follo ws. 1.16 Lemma Sup p ose that we ha v e a counta bly inﬁnite family of large-scale L ipsc hitz maps f k : X k ✲ Y k , with d ( f k ( u ) , f k ( v )) ≤ s k d ( u, v ) + t k . 8 Let s and t denote the n onstandard reals represented by the K -sequences ( s k ) and ( t k ) in ∗ R . Ass ume that s is ﬁnite with standard p art s td ( s ) = s ′ ∈ R . Su pp ose that r is a p ositiv e nonstandard real. If t r is in ﬁnitesimal, then f in duces an s ′ -Lipsc hitz map C ( f ) : C ( X D , p, r ) ✲ C ( Y D , f D ( p ) , r ) . This holds in particular if the s k and t k are constan t and r is any p ositive inﬁ nite n onstandard r eal. Pr o of. W e ha v e 1 r d ( f D ( u ) , f D ( v )) ≤ 1 r d ( u, v ) + t r b y Los’ Theorem. Identifying p oints at inﬁ nites- imal distance, w e obtain a w ell-deﬁned L ipsc hitz map b etw een the ultralimits. ✷ 1.17 C orollary There is n o quasi-isometric em b edding f : [0 , ∞ ) × R n ✲ R n . Pr o of. Other wise, w e could c ho ose an in ﬁnite p ositiv e nonstandard r eal r to obtain a conti nuous injection C ( f ) C (([0 , ∞ ) × R n ) D , 0 , r ) ⊂ C ( f ) ✲ C ( R n D , 0 , r ) [0 , ∞ ) × R n ∼ = ❄ R n . ∼ = ❄ Suc h a map would em b ed a closed n + 1-cub e homeomorphically into R n , whic h is imp ossible [17, 7.1.1,7 .3.20]. ✷ 2 Coarse equiv alences of trees In this section we p ro ve Theorem I from the in tro duction. W e ﬁrst in v estigate the structure of group actions on trees that are 2-transitive on the ends. 2.1 Deﬁnition A metric sp ace T is called a tr e e (or R -tree) if it h as the follo wing tw o p rop erties. (T1) F or an y tw o p oin ts x, y ∈ T , there is a u nique geo desic γ : [0 , d ( x, y )] ✲ T with γ (0) = x an d γ ( d ( x, y )) = y . W e pu t [ x, y ] = γ ([0 , d ( x, y )]). (T2) If 0 < r < s and if γ : [0 , s ] ✲ T is an injection suc h that γ | [0 ,r ] and γ | [ r,s ] are geo desics, then γ is a geo desic. T rees are CA T( κ ) for eve ry κ ∈ R , and in particular CA T(0) [7, I I.1.15]. Basic references for trees are [1], [14] and [33]. An R -tree with extensible geo desics is ca lled a le aﬂess tr e e . An ap artment in T is an isometric image of R (a geo desic line). If z is in the geodesic segmen t [ x, y ] but diﬀeren t from x and y , we sa y that z is b etwe en x and y . Giv en t wo geo desic segments [ x, y ] and [ x, y ′ ] in a tree, ther e is a unique p oint z with [ x, y ] ∩ [ x, y ′ ] = [ x, z ] [14, p . 30]. A p oint z in the tree is called a br anch p oint if there are thr ee p oint s u, v , w distinct from z such that [ u, v ] ∩ [ v , w ] ∩ [ w , u ] = { z } . 2.2 Ends A r ay in a tr ee T is an isometric im age of [0 , ∞ ) (a geo desic ra y). Two ra ys are call ed equiv alen t (or parallel) if their in tersection is again a r a y; the r esulting equiv alence classes are called the ends of T . T he s et of all ends of T is denoted by ∂ T . Give n x ∈ T and u ∈ ∂ T , there is a unique geodesic γ : [0 , ∞ ) ✲ T w ith γ (0) = x whose image is in the class of u [14, p. 60]; we pu t γ ([0 , ∞ )) = [ x, u ) and ( x, u ) = [ x, u ) − { x } . 9 Ev ery apartmen t in T d etermines tw o ends. Conv ersely , if u, v are distinct end s, then there is a unique apartmen t w hose end s are u and v [14, p. 61] and which w e denote ( u, v ) ⊆ T . If A ⊆ T is an apartmen t and z ∈ T , then there exists a uniqu e p oin t π A ( z ) in A which has min imal d istance from z and eve ry geodesic segment [ z , x ] with x ∈ A con tains π A ( z ) [14, p. 61]. Th is deﬁn ition is compatible with 1.1. W e n o w consider th e follo win g condition. (2- ∂ ) The group G acts isometrically on the lea ﬂess metrically complete tree T , and this act ion is 2-transitiv e on ∂ T : give n ends u 6 = v and u ′ 6 = v ′ , there is an elemen t g ∈ G with g ( u ) = u ′ and g ( v ) = v ′ . W e note that suc h a group a cts transitiv ely on t he set of apartmen ts of T . W e no w deriv e the structure of trees T s atisfying the condition (2- ∂ ). W e call a p oin t x in the tree G -isolate d if it is the un ique ﬁxed p oint of its stabilizer, T G x = { x } . F or s uc h a G -isolated p oint x , the stabilizer G x is by the Bruhat-Tits Fixed Po int T heorem [7, I I.2.8] a maximal b oun ded subgroup (i.e., G x is n ot prop erly con tained in a b ounded sub group of G ). 2.3 Prop osition Let u, v , w b e thr ee distinct end s of a tree T satisfying (2- ∂ ) and let x b e the branch p oint d etermin ed b y these three end s, ( u, v ) ∩ ( u, w ) = ( u, x ] . Then there exists an ele ment g ∈ G x whic h ﬁxes u and maps v to w . Pr o of. Sin ce G is 2-transitiv e on ∂ T , we ﬁ nd h ∈ G u suc h that h ( v ) = w . u v w h x h ( x ) Since h ( x ) ∈ h ( u, v ) = ( u, w ), one of the f ollo wing th ree cases m ust h old: (1) I f h ( x ) = x w e are done, with g = h . (2) If h ( x ) ∈ ( u, x ), th en A = S { h − n ( u, x ] | n ∈ N } is an h -stable apartment, on w hic h h indu ces a translation of length d ( x, h ( x )). As al l a partments are conjugate u nder G , w e ﬁ nd an isometry h ′ ﬁxing u and w whic h maps h ( x ) to x . Thus g = h ′ h ﬁ xes x and u and maps v to w . (3) I f x ∈ ( u, h ( x )), we p ut A = S { h n ( u, x ] | n ∈ N } and argue similarly as in (2). ✷ 2.4 Corollary In a tree satisfying (2- ∂ ), all b ranc h p oints are G -isolated. ✷ W e h a ve the f ollo wing a top ological d ic h otom y . 2.5 Prop osition In a tree satisfying (2- ∂ ), the set of branc h p oin ts is either closed and discrete, or it is dense in ev ery apartmen t. Pr o of. L et A = ( u, v ) b e an apartmen t and assume that the set of br anc h p oint s is not dense in A . If A con tains no b ranc h p oin t, then A = T , and if A con tains only one br anc h p oin t, then ev ery apartmen t con tains only one br anc h p oin t. In either case, the claim follo ws. Assume that x ∈ A is a br anc h p oint and that A con tains another branch p oint y . By 2.3 we ﬁnd an element g ∈ G x whic h interc hanges ( u, x ] and ( v , x ]. Then g ( y ) is also a branc h p oin t. In this 10 w a y , we get inﬁ nitely many bran c h p oint s in A at un iform d istance d ( x, y ). Because the set of branc h p oint s in A w as assumed n ot to b e dense, there has to b e a minimal distance t b et w een these, and they are distributed u niformly at this d istance in A . Since all apartments are conjugate, the set of b ranc h p oin ts is either dense or d iscrete with uniform distance t in eve ry apartment. In the discrete case, the m inimal d istance b et ween tw o branch p oint s is t , so th e set is closed and discrete in T . ✷ 2.6 Corollary Assume that the tree T satisﬁes (2- ∂ ) and that the set of branc h p oin ts is discrete. There are three p ossib ilities for the structure of T . T yp e (0) . There are n o branch p oin ts, T ∼ = R . T yp e (I) . There is a single branc h p oin t. T hen T is the Euclidean cone o v er its set of ends ∂ T , i.e. T is a quotien t of [0 , ∞ ) × ∂ T , wh er e 0 × v is identiﬁed with 0 × u , for all u, v ∈ ∂ T . T yp e (I I) . There is an inﬁnite discrete set of branc h p oints. Then T is a simplicial metric tree, ev er y v ertex has v alence at least 3 , and all edges h a ve the same length t . ✷ This classiﬁcation can b e reﬁned in terms of the G -action. F or t yp e (0), G induces a group {± 1 } ⋉ R on T , w here R is a su bgroup of ( R , +). If R 6 = 0, there are inﬁnitely man y G -isolated p oin ts. F or t y p e (I), there is a u nique G -isola ted p oin t. F or typ e (I I), G z acts transitiv ely on the set of edges con taining z , for eac h b ranc h p oint z . Hence there are tw o sub cases: either G acts transitiv ely on th e branc h p oin ts (v ertices), or T is bipartite and G has t wo orbits on the vertic es. I n the ﬁrst case, th e mid-p oints o f the edges are G -isolated (and G acts with inv ersion), in the second case, only the branch p oint s are G -isol ated. W e note that for the t yp es (I), (I I), T admits the structure of a simplicial metric tree with a simplicial G -action. W e say that T is of typ e (III) if the set of b ranc h p oin ts is dens e. By 2.5, the branc h p oin ts are then dense in ev ery apartment. 2.7 Prop osition Assum e that th e tree T satisﬁes (2- ∂ ) and that the set of branc h p oints is dense. Then every p oint x ∈ T is G -isolated. Pr o of. Let x ∈ T . If G x ﬁxes another p oin t y 6 = x , then G x ﬁxes the geodesic segment [ x, y ]. There is a b ranc h p oint z b et ween x and y . By 2 .3 y is not a ﬁxed p oint of G x,z . But G x,y = G x ⊇ G x,z , a con tradiction. ✷ Ev ery tr ee satisfying (2- ∂ ) corresp onds to one of the four t yp es (0)-(II I ). No w w e cla rify the role of the maximal b ounded subgroups. W e noted already that b y the Bruhat-Tits Fixed P oint Theorem, the stabilizers of G -isolated p oin ts are m aximal b ounded subgroup s. 2.8 Prop osition Assum e (2- ∂ ) an d that P ⊆ G is a maximal b ound ed subgroup. Then P is th e stabilizer of a G -isolated p oin t. Pr o of. Assum e this is f alse, so th e ﬁxed p oint set T P con tains a geo desic segment [ x, y ] with x 6 = y . If [ x, y ] co nta ins a branc h p oint z , then G z = P (b y maximalit y), whence T P = T G z = { z } , a con tradiction. F rom 2.5 we see that the set of br anc h p oint s is closed and d iscrete in T . By 2.6 we can assu me that T is a simplicial tr ee. Then P ﬁ xes some simp licial ed ge of T elemen t wise. Therefore it ﬁxes also some bran c h p oin t z , which is a con tradiction to 2.3. ✷ In a tr ee T satisfying (2- ∂ ), let i G ( T ) d enote the set of G -isolated p oin ts of T . By the results ab o v e, the set i G ( T ) corresp ond s bijectiv ely to the set of maximal b oun ded su bgroups of G . With resp ect to 11 the conju gation acti on of G on subgroups, this corresp ondence is G -equiv arian t. Ou r aim is to show that T can b e reco v ered from the G -actions on i G ( T ) and ∂ T . W e let b ( T ) ⊆ i G ( T ) denote the set of branc h p oin ts and consider the diﬀeren t t yp es (0)-(II I ) of trees. By the c ombinatorial structur e of a tree we mean the underlying set T together with the colle ction of all apartmen ts in T (without an y metric). First, we disp ose of the t wo degenerate t yp es (0) and (I). 2.9 Typ e (0) is charac terized by # ∂ T = 2 (and b ( T ) = ∅ ). Th e tree T ∼ = R is unique u p to isometry and the group indu ced by G splits as a semidirect p ro duct Z / 2 ⋉ R , wher e R is a subgroup of ( R , +). Since R con tains su bgroups whic h are abstractly , bu t not top ologica lly , isomorph ic, the G -actio n on T can in general n ot b e reco v ered from the G -action on i G ( T ) and ∂ T . T yp e (I) is charac terized b y # ∂ T ≥ 3 and # i G ( T ) = 1. Th is determines b oth the combinatoria l structure of the tree T and the G -actio n, but not th e metric. In the remaining cases, b oth ∂ T and i G ( T ) are inﬁn ite. T his situation is muc h more rigid. 2.10 Su pp ose th at T is of t yp e (I I). Then x ∈ i G ( T ) is a branch p oint if and only if G x has no orbit of length 2 in i G ( T ). So w e can reco v er the set b ( T ) of branch p oin ts in i G ( T ). By 2.3, tw o branc h p oint s x, y are adjacen t if and only if the only branch p oin ts ﬁxed b y G x,y are x and y , hence the simplicial structure of T can b e reco vered from the G -action on i G ( T ). W e n ote that then G x,y has at most 3 ﬁ xed p oin ts in i G ( T ). S ince all edges of T h a ve the same length, T is determined as a metric space up to a scaling factor. The follo wing result sho ws ho w f or type (I I I) apartmen ts can b e d escrib ed in terms of the G -action. Recall that π A : T ✲ A d enotes th e retraction that send s x to the nearest p oin t π A ( x ) ∈ A . 2.11 Lemma Assume that T is of t yp e (I I I). Let A = ( u, v ) b e an apartmen t. Th en x is con tained in A if and only if x is the unique ﬁxed p oin t of h G x,u ∪ G x,v i . Pr o of. S upp ose that x 6∈ A . Th en b oth G x,u and G x,v ﬁx π A ( x ) 6 = x . No w assume that x ∈ A and that z 6 = x . If π A ( z ) = x then x is a branch p oint an d G u,x mo v es z b y 2.3. If π A ( z ) is b etw een x and u , then there is a b ranc h p oin t b et we en x and π A ( z ) and therefore G v,x mo v es z b y 2.3. Similarly , G u,x mo v es z if π A ( z ) is b et ween x and v . ✷ 2.12 If T is of t yp e (I I I), th en i G ( T ) = T b y 2.7. Th e stabilizer G x,y of t w o distinct p oin ts x, y ﬁxes the inﬁn ite set [ x, y ] ⊆ i G ( T ), so typ e (I I I) can b e distinguished from type (I I). By 2.11 we can reco ver the apartmen ts in T fr om the group action. Let A = ( u, v ) b e an apartmen t and let z ∈ A . Then ( u, z ] = A ∩ T G u,z , so w e can also reco v er the rays in A . Therefore G determines the top olog y of A . Since the branc h p oin ts are dense A , the group G u,v induces a group H of translations on A whic h has a dens e orb it. Up to a scaling factor, there is ju st one H -in v ariant metric on A w hic h satisﬁes d ( h n ( x ) , y ) = nd ( x, y ) for all h ∈ H and all n ∈ N . So the metric on A is determined up to scaling. Since all apartments are conjugate, the m etric on T is un ique up to scali ng. Summarizing these facts we hav e the follo wing result. 2.13 Prop osition Assu me that T is a tree w ith # ∂ T ≥ 3 and that G acts on T in suc h a wa y that (2- ∂ ) holds. Giv en the G -a ctions on ∂ T and i G ( T ) , the com binatorial structure of the tree T is 12 uniquely determined. If T is of t yp e (I I) or type (I I I), the metric is determined up to a scaling factor. ✷ No w w e get to our main rigidity resu lt for trees. Supp ose th at f : T 1 ✲ T 2 is a coarse equiv alence b et wee n tw o trees. W e recall a result ab out coarse equiv alences b et we en δ - h yp erb olic spaces whic h go es bac k to M. Morse. It sa ys that the c oarse image of a geo desic is Hausdorﬀ close to a geo desic [7, I I I.H.1.7 ] [10, 1.3.2]. S ince trees are δ -hyperb olic for every δ > 0, it follo ws that there is a p ositiv e constant r f > 0 suc h that the image f ( A ) of an apartmen t A ⊆ T 1 has Hausdorﬀ distance at most r f from a unique apartmen t A ′ ⊆ T 2 . W e put f ∗ A = A ′ . Note that f ∗ just maps apartmen ts to apartments, it is not a map b et wee n th e trees. 2.14 F or our application to Euclidean bu ildings w e h a ve to consider the more general situatio n of a coarse equ iv alence f : T 1 × R m 1 ✲ T 2 × R m 2 . A leaﬂess R -tree is a 1-dimensional Euclidean b uilding, so we ma y apply the h igher dimensional Morse Lemma 5.9. It follo ws that there is a constan t r f > 0 su c h that for ev ery apartmen t A ⊆ T 1 there is a unique apartmen t A ′ ⊆ T 2 suc h that f ( A × R m 1 ) has Hausd orﬀ distance at most r f from A ′ × R m 2 . Moreo ver, m 1 = m 2 b y 5.13. W e deﬁn e a map f ∗ from the set of apartmen ts in T 1 to the set of apartmen ts in T 2 b y putting f ∗ A = A ′ . If g is a coarse inv erse for f , then g ∗ is an in v erse for f ∗ . W e also n eed the follo win g auxiliary resu lt on tr ees. 2.15 Lemma Let F b e a collection of apartments in a tree T and let r > 0 . If the subset X = T { B r ( A ) | A ∈ F } ⊆ T is nonempt y and un b ound ed, then the apartmen ts in F hav e a common end. Pr o of. T he result is a sp ecial case of 5.15 b elo w, but w e giv e a d irect pr o of. Let ( u, v ) ∈ F and c h o ose a sequence x n ∈ X su c h that π ( u,v ) ( x n ) conv erges to one end of ( u, v ), say u . This is p ossible since X ⊆ B r (( u, v )) is unboun ded. Let A ∈ F . Then d ( π A ( x n ) , π ( u,v ) ( x n )) ≤ 2 r and the unb ounded sequence π A ( x n ) sub conv erges to an end w of A . If w 6 = u , then the set { d ( π A ( x n ) , π ( u,v ) ( x n )) | n ∈ N } w ould b e unb ounded. Thus w = u . ✷ In the previous lemma the en d is unique, unless F consists of a single apartment. 2.16 Prop osition If T 1 has at least one branc h p oint and if f : T 1 × R m ✲ T 2 × R m is a coarse equiv alence, then the map f ∗ b et w een the sets of apartmen ts of the trees ind uces a bijection f ∗ : ∂ T 1 ✲ ∂ T 2 b et w een the ends of the trees, in su ch a wa y that f ∗ ( u, v ) = ( f ∗ u, f ∗ v ) . Pr o of. Let r f > 0 b e as in 2.14 and let F b e a ﬁn ite collection of apartmen ts in T 1 ha ving an end u in common. Put ( u, x ] = T { A | A ∈ F } and Y = T { B r f ( f ∗ A ) | A ∈ F } . Th en f (( u, x ] × R m ) ⊆ Y × R m . If th e set f ∗ F has no common end, th en Y is b ounded by 2.15 , so Y × R m is q uasi-isometric to R m . W e obtain then a qu asi-isometric em b eddin g of [0 , ∞ ) × R m in to R m , whic h is imp ossible 13 b y 1.17. Therefore Y is unb ounded and f ∗ F has a common end. Applying the s ame argumen t to a coarse inv erse g of f , w e see that a ﬁnite set F has an end in common if and only if f ∗ F has an en d in common. F or i = 1 , 2, consider the simplicial complex AC ( T i ) wh ose simplices are the ﬁnite collections of apartmen ts in T 1 ha ving a common end. The en ds of T i corresp ond to the m aximal complete sub complexes of AC ( T i ) (a s implicial complex is called complete if any tw o sim plices are contai ned in some simplex). W e ha ve shown that f ∗ induces an isomorphism f ∗ : AC ( T 1 ) ✲ AC ( T 2 ). It follo w s that f ∗ induces a bijection ∂ T 1 ✲ ∂ T 2 . ✷ The complex AC ( T i ) is a sp ecial case of the apartmen t complex of a spherical building; see 3.9. It is the n erv e of the co ve ring of ∂ T i giv en by all t wo-el ement subsets. The last paragraph of the pro of is a sp ecial case of 3.10 b elo w. The n ext result is a sp ecia l case of [25, 8.3.11]. 2.17 Prop osition Let G b e a group acting isometrically on t wo metrically complete leaﬂess trees T 1 , T 2 . Assu me that f : T 1 × R m ✲ T 2 × R m is a coarse equiv alence and that the induced map f ∗ on the apartments is G -equiv arian t. If T 1 has at least three ends, then a su bgroup P ⊆ G has a b ound ed orbit in T 1 if and only if it has a b ounded orbit in T 2 . Pr o of. S upp ose th at P ⊆ G has a boun ded orbit in T 1 . Let x b e a branc h p oint in T 1 and consider the b oun ded orbit P ( x ). W e put f ( x × 0) = y × q and w e show that y has a b ounded orbit P ( y ) in T 2 . Let F denote the set of all apartment s in T 1 whic h in tersect the orbit P ( x ) non trivially . Th is set F is obvio usly P -inv arian t and has no common end (b ecause x is a b ranc h p oint). Let F ′ = f ∗ F denote the corresp onding set of apartmen ts in T 2 . Since we assu me that f ∗ is G -equiv arian t, F ′ is also P -inv ariant, and th e apartments in F ′ ha v e n o common end by 2.16. F or s > 0 consider the P -in v arian t set X s = \ { B s ( A ′ ) | A ′ ∈ F ′ } ⊆ T 2 . By 2.15, the set X s is b ounded (or emp t y ). Let r ′ = sup { d ( x, p ( x )) | p ∈ P } . F or ev ery A ∈ F we ha v e d ( x, π A ( x )) < r ′ . Pu t f ( π A ( x ) × 0) = y ′ × p ′ . Then d ( y ′ , π f ∗ A ( y ′ )) ≤ r f b y 2.14. If ρ denotes the con tr ol fun ction for f , then d ( y , y ′ ) ≤ ρ ( r ′ ). This holds for all A ∈ F , whence y ∈ X ρ ( r ′ )+ r f . As this set is b ounded and P -inv ariant, P ( y ) is b ound ed. If g is a coarse in v erse of f , then g ∗ is G - equiv arian t (b ecause it is th e in v erse of the equiv arian t map f ∗ ), so w e obtain the conv erse imp licatio n b y the same arguments. ✷ No w we pro v e our main result on trees, which implies Theorem I in the introdu ction. The map f ∗ : ∂ T 1 ✲ ∂ T 2 is d eﬁned as in 2.16. 2.18 The orem Let G b e a group acting isometrically on t w o metrically complete leaﬂess trees T 1 , T 2 , with # ∂ T 1 ≥ 3 and assum e that the action of G on ∂ T 1 is 2 -transitive . Sup p ose that there is a coarse equiv alence f : T 1 × R m ✲ T 2 × R m and that the induced map f ∗ : ∂ T 1 ✲ ∂ T 2 is G -equiv ariant. T h en we hav e the follo w ing. 14 (i) After rescaling the metric on T 2 b y a constan t r > 0 , there is a G -equiv arian t isometry ¯ f : T 1 ✲ T 2 . F or ev ery apartment A ⊆ T 1 w e h a v e ¯ f ( A ) = f ∗ A . If T 1 has at least 2 branch p oints, then b oth ¯ f and r are unique. (ii) Put f ( x × p ) = f 1 ( x × p ) × f 2 ( x × p ) . If T 1 has at least 2 bran ch p oin ts, then there is a constan t s > 0 such that d ( f 1 ( x × p ) , ¯ f ( x )) ≤ s h olds for all x × p ∈ T 1 × R m . The constant s d ep ends only on T 1 , the control f u nction ρ , a and the constan t r f from 2.14. In p articular, f and ¯ f hav e ﬁnite distance if m = 0 . (iii) If f is a rough isometry (see 1.4), w e ma y put r = 1 . Pr o of. (i) By 2.17 b oth G -actions ha v e the same set of m aximal b oun ded subgroup s. These subgroups corresp ond by 2.8 to th e G -isolated p oin ts. Therefore we ha v e an equiv ariant b ijection ¯ f : i G ( T 1 ) ✲ i G ( T 2 ) . The t yp es (I), (II), and (I I I) can be distinguished b y the G -action on i G ( T 1 ); see 2.9, 2.10 and 2.12 (our assumptions exclud e trees of t yp e (0)). The com binatorial structure is also encod ed in the G -action, as we noted in 2.13. Also, w e can rescale th e m etric on T 2 b y a constan t r > 0 in s uc h a w ay that ¯ f : i G ( T 1 ) ✲ i G ( T 2 ) extends to an equiv ariant isometry T 1 ✲ T 2 whic h we al so den ote b y ¯ f . F rom the construction it is clear that ¯ f ( A ) = f ∗ A . F or trees of t yp e (I I) and (I I I), ¯ f and r are unique b y 2.13. (ii) Let z ∈ b ( T 1 ) and put r 1 = 1 + inf { d ( x, y ) | x, y ∈ b ( T 1 ) , x 6 = y } . Th en T 1 is cov ered b y the G -translates of B r 1 ( z ). Let F b e the collection of all apartments of T 1 con taining z . By 2.14 there is a constant r f > 0 s uc h that f 1 ( z × p ) ∈ T { B r f ( f ∗ A ) | A ∈ F } = T { B r f ( ¯ f ( A )) | A ∈ F } = B r f ( ¯ f ( z )). It f ollo ws that d ( f 1 ( x × p ) , ¯ f ( x )) ≤ r r 1 + r f + ρ ( r 1 ) for general x ∈ T 1 . (iii) F or trees of t yp e (I) it is clear that no rescaling is n ecessary in ord er to ﬁnd ¯ f . Supp ose that f is a r ough isometry , with con trol function ρ ( t ) = t + b . F or trees of t yp e (I I) and (I I I) w e ha ve b y (ii) d ( ¯ f ( x ) , ¯ f ( x ′ )) ≤ d ( x, x ′ ) + b + 2( r r 1 + r f + b ). On the other hand , d ( ¯ f ( x ) , ¯ f ( x ′ )) = r d ( x, x ′ ). Since T 1 is u n b oun ded, r ≤ 1. Applying the same argument to ¯ f − 1 , we see th at r = 1. ✷ F or the sp ecial case of a coarse equiv alence b et wee n locally ﬁnite simplicial trees, a similar result is pro v ed in [31, 4.3.1]. 3 Spherical buildings In this section we record some basic notions and facts ab out spherical bu ildings. E v erything w e need can b e foun d in [8], [38], [45] and [48]. F or our present pu rp oses it is con ve nient to view buildin gs as simplicial complexes. T his is essen tially T its’ appr oac h in [46]; see also [16]. 3.1 Simplicial complexes Let V b e a set and S a colle ction of ﬁn ite subsets of V . If S S = V and if S is closed under going do wn (i.e . a ⊆ b ∈ S implies a ∈ S ), then th e poset ( S, ⊆ ) is called a simplicial c omplex . Th e k + 1-elemen t sub sets a ∈ S are called k - simplic e s . More generally , any p oset isomorphic to suc h a p oset ( S, ⊆ ) will b e calle d a simplicial complex. The join S ∗ T of t w o simplicial complexes S, T is the pro du ct p oset; it is again a simplicial complex. Homomorphisms b et w een simplicial complexes are deﬁned in the ob vious w a y as order-preserving maps which d o not 15 raise the dimension of simplices [7, 7A.1]. A h omomorphism which m aps k -simplices to k -simplices is called non-de gener ate . The ge ometric r e alization | S | of the simplicial complex S is the set | S | =  p : V ✲ [0 , 1] | p − 1 (0 , 1] ∈ S and P v ∈ V p ( v ) = 1  Endo we d with the w eak top olog y , | S | is then a CW complex. Moreo v er, | S ∗ T | ∼ = | S | ∗ | T | , where the righ t-hand side is the t op ological join. In the case of spherical buildin gs, | S | is often endow ed with a stronger, metric top ology [7, I I.10A]. A result due to Do wke r sa ys th at th e iden tit y map is a homotop y equiv alence b etw een these t wo top olog ies [7, I.7]. 3.2 Coxeter groups and buildings Let ( W, I ) b e a C o xeter system [6, IV.3] [8, I I .4]. Thus W is a group with a (ﬁnite) generating set I consisting of inv olutions and a p resen tation of the form W = D I    ( ij ) or d ( ij ) = 1 for all i, j ∈ I E . F or a sub set J ⊆ I we p ut W J = h J i . If J is nonemp t y , then ( W J , J ) is again a Coxet er system [6, IV.8]. Th e p oset Σ = Σ( W , I ) = [ { W /W J | J ⊆ I } , ordered by rev er sed inclus ion, is a simplicial complex, the Coxeter c omplex [8, I I I.1]. T he typ e of a simplex w W J is t ( w W J ) = I − J . The t yp e function may b e view ed as a non-degenerate simplicial epimorphism from Σ to the p o wer set 2 I of I (view ed as a simplicial complex). A building B is a simplicia l co mplex tog ether with a collectio n Apt ( B ) of sub complexes, called ap artments , whic h are isomorphic to a ﬁxed Co xeter complex Σ. T he apartment s ha ve to satisfy the follo wing t w o compatibilit y conditions. (B1) F or an y t wo simp lices a, b ∈ B , there is an apartment A ∈ Apt ( B ) cont aining a, b . (B2) If A, A ′ ⊆ B are apartments con taining the simp lices a, b , then there is a (t yp e preserving) simplicial isomorphism A ✲ A ′ ﬁxing a and b . The t yp e functions of the apartment s are therefore compatible and ext end to a non-degenerate sur - jectiv e s implicial map t : B ✲ 2 I , th e type function of the b uilding. T he cardinalit y of I is th e r ank of the building, # I = rank( B ) = d im( B ) + 1 . A building of rank 1 is ju st a set (of cardinalit y at least 2 ), the apartmen ts are the t w o-elemen t subsets. The maximal simp lices in a bu ilding are called chamb ers . W e denote by C ham ( B ) the set of all c h am b ers. Every simplex of a buildin g is con tained in some c hamber (so b uildings are pur e simp licial complexes). Recall th at the dual gr aph of a pu re complex is the graph whose vertic es are th e maximal simplices and whose edges are the simp lices of codimension 1; this is the chamb er gr aph of ∆. A gal lery is a s implicial path in the c h am b er graph, and a nonstammering gallery is a path where consecutive c h am b ers are distinct. Th e c ham b er graph of a building is alw a ys connected. A minimal g al lery is a shortest path in the c ham b er graph. 16 A bu ilding is ca lled thick if every non-maximal simplex is con tained in at least 3 distinct cham b ers (it is alwa ys conta ined in at least 2 distinct c ham b er s). If every simp lex of co dimension 1 is con tained in exactly t w o c hambers, the building is called thin . Thin buildin gs are Coxe ter complexes. W e allo w non-thic k build ings (in [45], all b uildings are assumed to b e thick, bu t the results fr om [45] whic h we collect in this section h old f or n on-thic k buildings as w ell). 3.3 Residues and panels Let a ∈ B be simplex of type J . Th e r esidue of a is th e p oset Res( a ) = { b ∈ B | b ⊇ a } . If a is not a cham b er, then this p oset is again a b uilding, wh ose Coxet er complex is mo deled on W I − J [45, 3.12]. If a is a simp lex of co d imension 1 and typ e I − { j } , then Res( a ) is called a j -p anel . There is a n order-rev er sing p oset isomorphism b et wee n the simplicial complex B and the set o f all residues in B . Residues can also b e deﬁned in terms of the c ham b er graph, viewed as an edge-colored graph. Basica lly , this is a dictionary which all o ws the passage from buildin gs, view ed as simp licial complexes (as in [45]) to buildings view ed as edge colo red graphs (or c ham b er systems) (as in [48]). In view of this corresp ondence, we call a simplex of type I − { j } also a j -panel. The join of t w o buildin gs is again a building. C on v ersely , a bu ilding decomposes as a join if it s Co xeter group is decomp osable (i.e. if I ⊆ W d ecomp oses in to t wo su bsets w hic h cen tralize eac h other). 3.4 Spherical buildings A Co xeter complex Σ is called spheric al if it is ﬁnite. Then the geometric realizatio n | Σ | is a com binatorial sph ere of dimension # I − 1. A bu ilding is called spheric al if its apartmen ts are ﬁnite (the buildin g ma y neve rtheless b e inﬁnite). Tw o simplices a, b in a sph erical Co xeter complex Σ are called opp osite if they are interc hanged b y the an tip o dal map (the opp osition in v olution) of the sphere | Σ | [45, 3.22]. In a spherical buildin g, t wo simplices are called opp osite if they are opp osite in some (h ence ev ery) apartmen t cont aining them. 3.5 Thick reduc tions A non-thic k spher ical building B can alw a ys b e ‘reduced’ to a thic k b uilding as follo ws. There exists a thic k sph erical buildin g B 0 suc h that B is a simplicial r eﬁnemen t of a j oin S 0 ∗ · · · ∗ S 0 ∗ B 0 (w e view S 0 as a thin sp herical buildin g of rank 1) [11] [13] [25, 3.7] [27, 3.8] [42]. F or the geometric realization, we hav e then | B | = S k ∗ | B 0 | , where k is th e num b er of S 0 -factors in the join. So non-thic k spherical bu ildings are susp ensions of thic k spherical buildin gs. The geometric realiza tion of a th in spherical building is a sph ere. The follo win g lemma w ill b e used later. 3.6 Lemma Let B b e a spherical building and let A ⊆ B b e an apartmen t. If ev ery panel a ∈ A is con tained in at least three diﬀerent cham b er s of B , then B is thic k. Pr o of. Let a b e an arbitrary p anel in B , and let c 0 , · · · , c k b e a s hortest gallery with th e prop er t y that the ﬁrst c hamber c 0 con tains a and the last c ham b er c k is in A . This gallery can b e con tinued inside A as a min imal gallery until it reac hes a p anel b ∈ A wh ic h is opp osite a . Then th ere is a bijection [ a ; b ] b et w een Res( a ) and Res( b ); see 3.12 below. It follo ws that a is con tained in at least three diﬀerent c h am b ers. ✷ 17 W e will see that Euclidean buildings give rise to a family of building epimorphisms. The follo wing fact is useful; see [5, 2.8]. 3.7 Lemma Let B , B ′ b e spherical bu ildings of the same t yp e and let ϕ : B ✲ B ′ b e a typ e- preserving simplicial map. Th en ϕ is sur jectiv e if and only if its restriction to ev ery panel is su rjectiv e. Pr o of. Since the c ham b er graph of B ′ is connected, it is clear that the lo cal surjectivit y condition implies global surj ectivit y . Con v ersely , su pp ose that ϕ is s urjectiv e. Let a ′ , b ′ ∈ B ′ b e i -adjacen t c h am b ers and let a b e a pr eimage of a ′ . W e ha v e to ﬁnd a preimage b of b ′ whic h is i -adjacen t to a . Let c ′ b e opp osite a ′ and let a ′ i b ′ · · · c ′ b e a minimal gal lery wh ic h w e denote b y γ ′ . Let c b e a preimage of c ′ . Since ϕ do es not increase distances in the cham b er graph , c is opp osite a . Thus there is a gallery a i b · · · c in B of the same t yp e as γ ′ , whic h w e denote b y γ . Since γ ′ is the unique gallery of its t yp e in B ′ from a ′ to c ′ , it follo ws that ϕ ( γ ) = γ ′ . In particular, ϕ ( b ) = b ′ . ✷ 3.8 Corollary Let B , B ′ b e sp h erical buildings of the same typ e and let ϕ : B ✲ B ′ b e a t yp e- preserving simplicial surj ectiv e map. If B ′ is thic k, then B is also thic k. ✷ A thick sph erical build ing is determined by its apartmen t complex whic h w e introdu ce now. This complex app eared already in 2.16 for the sp ecial case of rank 1 b uildings. W e will see later that the apartmen t complex of the spherical bu ilding at inﬁ nit y is a coarse inv arian t of a Euclidean buildin g. 3.9 The apartment complex Let B b e a buildin g and Apt ( B ) its set of apartment s. Th e ap artment c omplex AC ( B ) is the simplicial complex w hose simp lices are ﬁnite subsets A 1 , . . . , A k of apartments, with A 1 ∩ · · · ∩ A k 6 = ∅ (in other words, AC ( B ) is th e nerve of th e co v ering Apt ( B ) of B ). If B is thic k, then every simp lex a ∈ B can b e wr itten as an inte rsection of ﬁnitely m an y apartment s. 3.10 Prop osition Let B 1 , B 2 b e thic k sph erical buildings and let ϕ : AC ( B 1 ) ✲ AC ( B 2 ) b e a simplicial isomorphism. T hen there is a uniqu e simplicial isomorphism Φ : B 1 ✲ B 2 suc h that ϕ ( A ) = Φ( A ) for all A ∈ Apt ( B 1 ) . Pr o of. Th e map ϕ sends sets of apartments with the ﬁn ite in tersection prop er t y to s ets with the ﬁnite in tersection pr op ert y . Because B 1 is thic k and has ﬁnite apartments, the maximal su bsets with the ﬁnite in tersection prop ert y in Ap t ( B 1 ) are precisely the sets S v = { A ∈ Apt ( B 1 ) | v ∈ A } for all v ertices v of B . T herefore ϕ induces a b ijection Φ b etw een th e v ertices of the buildin gs. If v ∈ A , th en Φ( v ) ∈ ϕ ( A ). S ince B 1 is th ic k, t wo ve rtices u, v in B 1 are adjacent if and only if the f ollo wing h olds: no other vertex w is in the in tersection of all apartmen ts cont aining u and v . Th is can b e expressed in AC ( B 1 ) as follo w s: if S w ⊇ S u ∩ S v , then S w = S u or S w = S v . S o Φ preserve s the 1-skel eton of B 1 . But ev ery b uilding is the ﬂag complex of its 1-sk eleton [45, 3.16 ], therefore Φ is simp licial. ✷ In the previous pro of, thickness is essen tial. It is clear that the th ic k reduction (see 3.5) of a spher- ical building has the same apartment complex as the bu ilding itself. W e remark that a simplicial isomorphism b et ween tw o spherical buildings maps apartmen ts to apartments, even if it is not typ e- preserving. 18 3.11 Pro jections in buildings Let c b e a c hamber and a a simplex in a building. Then there is a u nique c hamb er d in Res( a ) w hic h has m inimal distance fr om c (with r esp ect to the distance in the c ham b er graph), and whic h is d enoted d = p ro j a c . If b is a simplex then p ro j a b is deﬁned to b e the simplicial in tersection of the cham b ers pro j a c , wh ere c ranges o v er all c ham b ers con taining b [45, 3.19]. If a, b are opp osite simplices in a s pherical building, then pr o j a : Res( b ) ✲ Res( a ) is a simplicial isomorphism [45, 3.28] . The follo wing observ ations are due to Knarr [26] and Tits [4 6 ]. They w ere redisco v ered by Leeb [31, C h. 3]. 3.12 P ersp ectivities Let B b e a s pherical buildin g. W e n oted already that if a, b are op p osite simplices in B , then pr o j b : Res( a ) ✲ Res( b ) is a building isomorphism (not necessarily typ e preserving) b etw een Res( a ) and Res( b ). W e d enote th is isomorphism by [ b ; a ] : Res( a ) ✲ Res( b ) and call it a p ersp e ctivity . A concatenation of p ersp ectivities is call ed a pr oje ctiv ity ; we wr ite [ c ; b ] ◦ [ b ; a ] = [ c ; b ; a ] : Res( a ) ✲ Res( c ) etc. Th e in v erse of [ b ; a ] is [ a ; b ]. A pro jectivit y is called even if it can b e written as a comp osition of an even num b er of p er sp ectivities. 3.13 Pro jectivities Recall that a gr oup oid is a small cate gory where every arro w is an isomorph ism. The pr oje ctivity gr oup oid Π B of a sph erical b uilding B is the category wh ose ob jects are th e simp lices of B , and whose morp hisms are pro jectivities. It is closely related to the opp osition gr aph Opp( B ) whose v ertices are the simplices of B and whose edge s are unordered pairs of opp osite simplice s. Ev ery simplicial path in Op p( B ) ind uces a pr o jectivit y . W e denote by Π B ( a ) = Hom Π B ( a, a ) the group of all automorphisms of Res( a ) induced b y maps in Π B . Th e subgroup Π B ( a ) + consisting of all eve n p ro jectivitie s is a normal sub group of index 1 or 2 in Π B ( a ). If f : B 1 ✲ B 2 is an isomorphism of sp herical buildings, then f indu ces an isomorphism b et wee n Π B 1 and Π B 2 in the ob vious wa y . The follo wing result is essentia lly Kn arr’s [26, 1.2]. W e use several facts ab out galleries and distances whic h can all b e found in [48]. 3.14 The orem Supp ose that B is a thic k spherical b uilding and that r is an i -panel. If i is n ot an isola ted no de in the Co xeter diagram of B , then Π B ( r ) + is a 2 -transitiv e p ermutation group on R = Res( r ) . Pr o of. Let a, b, b ′ b e thr ee distinct c ham b ers in R . W e construct a pro jectivit y which ﬁxes a and maps b to b ′ . Let j d enote the type of a n eigh b oring nod e of i , i.e. ij 6 = j i in W . W e c ho ose a nonstammering galle ry b i a j c i d in B (the sup erscripts indicate the t yp es of p anels in th e gallery). Since ij 6 = j i , this gallery is minimal. Th erefore it is con tained in some apartment A ⊆ B . Let s b e the panel in A opp osite to the j -panel a ∩ c . Let e b e a c hamber in S = Res( s ) w hic h is n ot in A (here we u se that B is thic k). Then e is opp osite to a and c . Th ere is a un ique p anel t ⊆ e 19 whic h is opp osite b oth to r and to the panel q = c ∩ d . Since b is n ot opp osite e , p ro j t b = e . Sim ilarly pro j t d = e , whence [ q ; t ; r ]( b ) = d . W e claim that pr o j t ( a ) = pro j t ( c ). Assumin g for the m omen t that this is true, we h a ve [ t ; r ]( a ) = [ t ; q ]( c ), whence [ q ; t ; r ]( a ) = c . Applyin g the same constru ction to b ′ i a j c i d , with a second apartmen t A ′ and a panel t ′ , w e obtain the pro jectivit y [ r ; t ′ ; q ; t ; r ] with the required pr op erties. It remains to sho w that pr o j t ( a ) = pro j t ( c ). Let A ′′ denote the apartment sp anned by the opp osite c h am b ers a and e . Let f d enote the c hamber in Res( t ) ∩ A ′′ diﬀeren t from e and let k denote the gallery distance b et w een a and e . Then a and f ha v e ga llery distance k − 1. S ince t and p are not opp osite, g = p ro j p f has gallery distance k − 2 from f . So there is a gallery ( f , . . . , g , a ) of length k − 1, whic h is therefore minimal. It follo ws that ( f , . . . , g , c ) is also a minimal gallery , and f has gallery distance k − 1 from c . Since t is opp osite to r and q , we hav e p ro j t ( a ) = f = p ro j t ( c ). ✷ Let a, b b e opp osite panels in B and let B ( a, b ) denote the union of all apartmen ts con taining a and b . F or a cham b er c ∈ B , let c a = pro j a c and c b = pro j b c . If A is an apartment contai ning a and b and c ∈ A is a cham b er, th en c a , c b ∈ A and p ro j a c b = c a . In p articular, eac h pair of distinct c ham b er s c, d ∈ Res( a ) d etermines a uniqu e apartmen t in B ( a, b ) conta ining c and d and the cham b ers in Res( a ) corresp ond bijectiv ely to the half apartments of B ( a, b ) h a vin g a and b as b ou ndary panels. 3.15 Lemma The sub complex B ( a, b ) ⊆ B is a w eak building of th e same type as B and ev ery apartmen t of this w eak building conta ins a and b . If b ′ is another panel op p osite a , then there is a unique simplicial isomorphism B ( a, b ) ✲ B ( a, b ′ ) whic h ﬁxes B ( a, b ) ∩ B ( a, b ′ ) . Pr o of. Let c, d b e cham b ers in B ( a, b ). Then there exists an apartment A conta ining a, b, c a and d a . It follo ws that A con tains c b and d b and therefore c and d . T his shows that B ( a, b ) is a b uilding and that ev ery apartment of th is b uilding conta ins a and b . Supp ose no w that b ′ is also opp osite a . F or ev ery apartment A con taining a, b , there is a unique apartmen t A ′ con taining a, b ′ , suc h that A ∩ Res( a ) = A ′ ∩ Res ( a ). Sin ce A ∩ A ′ con tains c hambers, there is a un ique isomorp hism A ✲ A ′ ﬁxing A ∩ A ′ . F or c ∈ A , the image c ′ ∈ A ′ can b e describ ed as follo ws. It is the unique c hamber wh ic h has the same W -v alued distances from the t wo cham b ers a c and pro j b ′ a c as c has from a c and pro j b a c = b c . This descrip tion is indep enden t of A and A ′ and sho ws that we obtain a wel l-deﬁned map on the c ham b ers. This map is adjacency-preserving on eac h apartmen t and hence a bu ilding isomorphism. ✷ 3.16 Su pp ose that a = a 0 is a panel and a 0 a 1 · · · a k is a path in the opp osition graph. By the previous lemma we obtain a sequence of canonical isomorph isms B ( a 0 , a 1 ) ∼ = B ( a 1 , a 2 ) ∼ = . . . ∼ = B ( a k − 1 , a k ) ﬁ xing th e intersecti ons of consecutive b uildings. If a 0 = a k , th en th e comp ositio n of these isomorphisms yields an automorphism of B ( a 0 , a 1 ). If k is even, then this automorphism ﬁxes a and its restriction to Res( a ) coincides with th e pr o jectivit y [ a k ; . . . ; a 0 ]. 4 Euclidean buildings The notion of a (nondiscrete) Eu clidean building is due to Tits [46]. Prior to their axiomatiz ation in [46], the nondiscrete Euclidean buildin gs that arise from reductiv e groups o ver v alued ﬁelds were studied in [9 ]. W e rely on Parreau’s w ork [35] whic h conta ins many imp ortan t structural r esults for Eu clidean buildings. She show ed in particular that the axioms giv en b y Kleiner-Leeb [25] are equiv alen t to Tits’ original axioms plu s m etric completeness. 20 4.1 The aﬃne W eyl group W e ﬁx a sp herical Co xeter group ( W , I ) in its stand ard representa tion on R n (where n = # I ). A Weyl chamb er or se ctor in R n is the closure of a connected comp onent in R n − ( H 1 ∪ H 2 ∪ · · · ∪ H r ), where the H k are the reﬂection hyp erplanes of W corresp onding to the conjugates of the generato rs i ∈ I . The clo sure of a connected comp onent of R n − H k is called a half sp ac e . A Weyl simplex in R n is an int ersection of W eyl c ham b ers. A wal l is a r eﬂection hyp erplane. W eyl cham ber half s pace W eyl simplex wall Note that we d o not require th at W b e irr educible. The W eyl simplices in R n , ord ered by inclusion, form a simplicial complex wh ic h is isomorphic to the Co xeter complex Σ of W . W e also ﬁ x a W -inv ariant inner p ro duct on R n . Th e corresp onding norm will b e d enoted by || · || . Up to scaling factors on the irr educible W -submo d ules of R n , suc h an inner pro du ct is uniqu e. The group W norm alizes the translation group ( R n , +) and the semidirect p ro duct W R n acts isometrically on R n . W e call this group W R n the aﬃne Weyl gr oup . 4.2 Euclidean buildings Let W b e a sph erical C o xeter group and W R n the corresp ond ing aﬃn e W eyl group. Let X b e a metric space. A chart is an isometric em b edding ϕ : R n ✲ X , and its image is called an aﬃne ap artment . W e call tw o c harts ϕ, ψ W -c omp atible if Y = ϕ − 1 ψ ( R n ) is con v ex (in the Euclidean s ense) and if ther e is an elemen t w ∈ W R n suc h that ψ ◦ w | Y = ϕ | Y (this cond ition is v oid if Y = ∅ ). W e call a metric s pace X toge ther with a collection A of charts a Euclide an b uilding if it has the f ollo wing prop erties. (EB1) F or all ϕ ∈ A and w ∈ W R n , the comp osition ϕ ◦ w is in A . (EB2) Any t w o p oin ts x, y ∈ X are con tained in some aﬃne apartment. (EB3) Th e charts are W -compatible. The charts allo w us to m ap W eyl cham b ers, walls and half spaces in to X ; their images are also called W eyl c hambers , w alls and half spaces. The ﬁr st thr ee axioms guarantee that th ese notions are co ordinate indep enden t. W e call A an atlas or ap artment system for X . (EB4) If a, b ⊆ X are W eyl cham b ers, th en th ere is an aﬃne apartment A such that the in tersections A ∩ a and A ∩ b con tain W eyl cham b er s. (EB5) If A 1 , A 2 , A 3 are aﬃne apartments whic h intersec t pairwise in half s paces, then A 1 ∩ A 2 ∩ A 3 6 = ∅ . The last axiom ma y b e replaced by the f ollo wing axiom [35, Th m. 1.21]. (EB5’) If A ⊆ X is an aﬃne apartmen t and x ∈ A a p oint, then there is a 1-Lipschitz retractio n ρ : X ✲ A with ρ − 1 ( x ) = { x } . The p oint x is calle d the c enter of the retraction. In fact, the retractions can b e c hosen in su c h a w a y that they ha v e th e follo wing slight ly stronger prop erty . (EB5 + ) I f A ⊆ X is an aﬃne ap artmen t and x ∈ A a p oin t, then there is a 1-Lipsc hitz retraction ρ : X ✲ A with d ( x, y ) = d ( x, ρ ( y )) for all y ∈ X . 21 The n umb er n is called the dimension of the Euclidean bu ilding X . It coi ncides with the top ological (co vering) dimen sion of X [28, 7.1] [30, 3.3 ]. In [46 ], [25] or [ 35], the trans lation group of the aﬃne W e yl group ma y b e some W -inv ariant subgroup of ( R n , +). Sin ce w e are in this p ap er only concerned with metric prop erties o f Euclidean buildings, and since the aﬃne W e yl group can alw a ys b e enlarged to th e full translation group without c han ging the underlying metric space and th e set of aﬃne apartments [35, 1.2], there is no loss in generalit y here. F rom ou r viewp oint, ev ery p oin t p ∈ X is a sp e cial p oint [9, 1.3.7]. W e r emark that the C o xeter group W of a Euclidean building need not b e crys tallog raphic [21] [4]. 4.3 Example: Euclidean cones o ver spherical buildings Let B b e a spherical building and let E B( B ) den ote the quotien t of | B | × [0 , ∞ ) wh ere | B | × 0 is id en tiﬁed to a p oin t. Let d | B | denote the sph erical metric on | B | , and put d ( x × s, y × t ) 2 = s 2 + t 2 − 2 st cos( d | B | ( x, y )); see [7, I.5.6]. With this metric, E B( B ) is the inﬁn ite Euclidean cone o ver | B | . It is not diﬃcult to see that EB( B ) is a Euclidean buildin g. Th e aﬃne apartmen ts in EB( B ) corresp ond bijectiv ely to the apartment s of B . Th ese buildings are generalizatio ns of the trees of typ e (I), and w e call them Eu c lide an buildings of typ e (I) . On e ca n view EB( B ) as the aﬃne building with resp ect to th e trivial v aluation on the spherical bu ilding B . W e note that th is constru ction is fun ctorial: every automorph ism of B exte nds to an isometry of EB( B ). In th is wa y , ev ery sph erical b uilding can b e view ed as a Euclidean b uilding. In [39], EB( B ) is calle d an immeub le ve ctoriel . 4.4 Example: leaﬂess trees A leaﬂess tree is a 1-dimensional Euclidean buildin g. In particular R is a Euclidean bu ilding. T he aﬃn e W eyl group is W R = { x 7− → ± x + c | c ∈ R } . 4.5 Example: simplicial Euclidean buildings The geometric realiza tion of an aﬃne s implicial building is a Euclidean bu ilding [8] [46]. The images of the W eyl simplices in a Euclidean building X un der the c harts are also called W eyl simplices. Th e image of the origin in R n is called the b ase p oint or tip of the W eyl simplex. 4.6 The v ector distance Let a 0 ⊆ R n b e a ﬁ xed W eyl c ham b er. Giv en tw o p oin ts p, v in the Euclidean build ing X , th ere exists a c hart ϕ : R n ✲ X that maps a 0 to a p -based W eyl c hamber con taining v . Let Θ( p, v ) ∈ a 0 ⊆ R n denote the ve ctor that is mapp ed to v . By (EB3 ), th e v ector Θ( p, v ) is indep endent of ϕ . W e thus hav e a w ell-deﬁned m ap X × X ✲ a 0 , ( p, v ) 7− → Θ( p, v ) whic h we call the ve ctor distanc e [35, 1.3.1 ]. W e remark that the map Θ( p, − ) : X ✲ a 0 is 1-Lipsc h itz. W e also note th e follo wing. The inv olution x 7− → − x on R n induces an inv olution j : x 7− → w 0 ( − x ) on the W eyl c ham b er a 0 , where w 0 is the unique longest eleme nt in the sph erical Co xeter group W . W e ha ve the symmetry r elation Θ( x, y ) = j (Θ( y , x )) , for all x, y ∈ X . The aﬃne apartmen t X = R n with A = W R n is an example of a Eu clidean building and we h a ve the follo wing small fact which will b e needed later. A related resu lt is pro ved in [35, 2.15]. 22 4.7 Lemma Let W b e a spherical Co xeter group, acting in the standard r epresen tation on R n . Let Θ denote the corresp onding vect or distance on R n . S upp ose that K ⊆ R n is a nonempt y conv ex set, that g : K ✲ K is a b ijection, and that Θ( p, v ) = Θ( g ( p ) , g ( v )) holds for all p, v ∈ K . T h en there exists an elemen t w in the aﬃn e W eyl group W R n whose restriction to K coincides w ith g . Pr o of. First we note that g is an isometry of K , b ecause || Θ( p, v ) || = d ( p, v ). W e c ho ose the pair p, v ∈ K in su c h a wa y that the smallest W eyl simplex b ⊆ a 0 that con tains Θ( p, v ) has maximal dimension. App lying an element of the aﬃne W eyl group W R n to K , we ma y assu me that p = 0 an d that v ∈ b ⊆ a 0 . W e choose w ∈ W R n in such a w a y th at w (0) = g (0) and w ( v ) = g ( v ). W e claim that g ( x ) = w ( x ) for all x ∈ K . In general, if Θ(0 , u ) = Θ(0 , u ′ ), then u and u ′ lie in the same ﬁnite W -orbit in R n . Since b is maximal, v is an in terior p oin t of the W eyl simplex b an d there is an ε > 0 suc h that g ( u ) ∈ w ( b ) holds for all u ∈ b ∩ K with d ( u, v ) ≤ ε . This in turn im plies that g ( u ) = w ( u ) for all u ∈ b ∩ K with d ( u, v ) ≤ ε . S ince b oth g and w are isometries, w e conclude that g and w agree on the con v ex set b ∩ K . Let now u ∈ K b e arbitrarily . If t > 0 is small we ha ve necessarily z = ut + v (1 − t ) ∈ b b y the maximalit y of b . Since b oth g and w are isometries and g ( v ) = w ( v ) and g ( z ) = w ( z ), we ha ve g ( u ) = w ( u ). ✷ The W eyl sim plices lead to t wo spherical b uildings. On e of them captures the asym ptotic geometry of X , while the other is an ‘inﬁn itesimal’ version of the Euclidean buildin g, similar to the tangent space of a Riemannian manifold. T he ﬁrs t one, th e spherical b uilding at inﬁ nit y , will b e considered no w and the second in 4.14. 4.8 The spherical building at inﬁnit y W e call t wo W eyl simplices a, a ′ ⊆ X Hausdor ﬀ e quivalent if they ha ve ﬁn ite Hausdorﬀ distance. The equiv alence class of a is denoted ∂ a . The preorder ⊆ H d deﬁned in 1.6 ind uces a partial order on these equiv alence classes. Let ∂ A X denote the set of all equiv alence c lasses o f W eyl simplices, partially ordered b y domination ⊆ H d . F or ev ery aﬃne apartmen t A , the p oset ∂ A consisting of the W eyl simplices conta ined in A ma y b e view ed as a sub-p oset of ∂ A X . 4.9 Prop osition The p oset ∂ A X is a spherical b uilding. The map A 7− → ∂ A is a one-to -one corre- sp onden ce b et we en the aﬃne apartments in X and the apartments of the spherical bu ilding ∂ A X . Pr o of. See Parreau [35 , 1.5]. ✷ W e h a ve also the f ollo wing fact. 4.10 Lemma Sup p ose that a and a ′ are W eyl cham b ers with tips x, x ′ . If ∂ a = ∂ a ′ , then H d ( a, a ′ ) ≤ d ( x, x ′ ) . Pr o of. L et ψ : a ✲ a ′ denote the unique Θ-inv ariant isometry . Giv en z ∈ a , there is a u nique geod esic γ : [0 , ∞ ) ✲ a with γ (0) = x and γ ( s ) = z , for some s ≥ 0. Th e geod esic γ ′ = ψ ◦ γ is at ﬁnite d istance from γ , b ecause ∂ a = ∂ a ′ . S ince the metric of a C A T(0) space is conv ex, w e ha v e d ( γ ( s ) , γ ′ ( s )) ≤ d ( x, x ′ ) [7, I I.2.2]. ✷ 23 4.11 The maximal atlas T he spherical building at inﬁnit y dep ends v ery muc h on the c hosen set of c h arts A . Similarly to a diﬀerent iable m anifold, a E uclidean buildin g admits a u nique maximal atlas ˆ A [35, 2.6]. The set ˆ A is called the c omplete ap artment system . W e denote the spherical building at inﬁnity wh ic h corresp onds to the c omplete apartmen t system by ∂ cpl X . Its metric realization coincides with the Tits b oundary of X [35, Cor. 2.19]. 4.12 Pro duct decomp ositions and reductions If the Co x eter group W is redu cible, then X decomp oses as a m etric pro d uct X = X 1 × X 2 of Euclidean buildings, with ∂ A 1 X 1 ∗ ∂ A 2 X 2 = ∂ A X [35, 2.1]. If S k ∗ B 0 = ∂ A X is the thick redu ction of ∂ A X and if W 0 is th e Coxe ter group of B 0 , then th ere is a Euclidean building X 0 with aﬃn e W eyl group W 0 R m and atlas A 0 and an isometry X ∼ = R n − m × X 0 , with ∂ A 0 X 0 = B 0 and k = n − m − 1 [25, 4.9]. W e record some more results ab out Eu clidean buildings whic h will b e needed later. 4.13 Prop osition Let X b e an n -dimensional Euclidean bu ilding. If A ⊆ X is isometric to R n , then A is an apartment in the complete apartment system of X . Pr o of. See [35, Prop. 2.25]. ✷ W e r emark that the p revious pr op osition is in the app roac h by Kleiner and Leeb essentiall y an axiom. 4.14 The residue at a p oin t Let p b e a p oin t in X . Two p -b ased W eyl simplices a, b ha ve e qual germs ne ar p if B r ( p ) ∩ a = B r ( p ) ∩ b holds for some r > 0. The corresp onding equiv alence classes form a s pherical building X p whic h w e call the r esidue of X at p ; see [35, 1.6]. T he Co xeter group of X p is W , hence X p has the s ame t yp e as the sp herical b uilding at inﬁnity ∂ A X . W e call the p oint p thick if the resid ue X p is th ic k. If a ⊆ X is a W eyl simplex, then ∂ a has a unique r epresen tativ e w hic h is a p -based W eyl simplex [35, Cor. 1.9] . The apartmen ts of X p arise from the aﬃne apartmen ts conta ining p (but this corresp ondence is in general not 1-to-1). W e th us obtain a t yp e-preservin g surjectiv e simplicial map ∂ A X ✲ X p , and also a surj ectiv e canonical 1-Lipschitz map EB( ∂ A X ) ✲ X (whic h dep end s on p ). 4.15 Giv en a p -based W eyl c h am b er a , let ζ a denote the s pherical barycen ter of the W eyl c ham b er ∂ a . W e call the geo desic ray [ p, ζ a ) ⊆ a the c entr al r ay of a . central r ay The p ossible angles b et we en all centrals ra ys starting at x form a ﬁ nite su bset of [0 , π ], since th e spherical Coxe ter complex of W is ﬁnite. If [ p, ζ b ) and [ p, ζ c ) are the cen tral ra ys of tw o p -b ased W eyl c h am b ers b, c , then the Alexandro v angle b et we en these ra ys is 0 if and only if b and c ha v e the same germ near p . 24 If t w o su ﬃcien tly long geo desic segments in a tree hav e s mall Hausdorﬀ distance, then th ey in tersect in a long geo desic segmen t. The next pr op osition is the higher d imensional analog of this fact. 4.16 Prop osition Let X b e a Eu clidean b uilding and let s > 0 be a p ositiv e real n umber . Th ere exists a p ositiv e real num b er c s > 0 s uc h that the follo wing holds for all t > 0 . If A, A ′ ⊆ X are aﬃne apartmen ts and y is a p oin t in X suc h that H d B t + c s ( y ) ( A, A ′ ) ≤ s , then B t ( y ) ∩ A = B t ( y ) ∩ A ′ . Pr o of. Let a 0 ⊆ R n denote the standard W eyl c ham b er. There exists a num b er c s > 0 s uc h that for the p oint z a ∈ [0 , ζ a ) with d (0 , z a ) = c s , the closed ball ¯ B 2 s ( z a ) ∩ a con tains only interior p oints of a . Supp ose n o w that t > 0, that A, A ′ ⊆ X are as in the claim of the prop osition and that p ∈ B t ( y ) ∩ A . W e claim that p ∈ B t ( y ) ∩ A ′ . W e choose a p -based W eyl c h am b er b ⊆ A and z b ∈ [ p, ζ p ) with d ( p, z b ) = c s . Th e condition on th e Hausdorﬀ distance ensures that we can ﬁ nd a p oint v ′ ∈ A ′ with d ( z b , v ′ ) ≤ 2 s . F rom the comparison triangle in Euclidean s pace w e see that the angle b et w een the geo desic segmen ts [ p, v ′ ] and [ p, z b ] is so small that [ p, v ′ ] in tersects the W eyl c ham b er b in an in terior p oint. p v ′ [ p, ζ b ) b ′ b W e n o w c ho ose a geodesic ra y [ v ′ , η ) ⊆ A ′ whic h extends [ p, v ′ ]. Th us [ p, v ′ ] ∪ [ v ′ , η ) is a geod esic ray . This ra y is cont ained in a unique p -based W eyl cham b er b ′ whose germ n ear p is con tained in A . W e n o w consider the p -based W eyl c h am b er c ⊆ A opp osite b , and we c ho ose z c ∈ [ p, ζ c ) similarly as b efore, w ith d ( p, ζ c ) = c s . Then we ﬁ nd a p -based W eyl cham b er c ′ whose germ n ear p is opp osite to b ′ . Th ese tw o W eyl c ham b ers are contai ned in a u nique aﬃne apartmen t [35, Pr op. 1.12], whic h is therefore A ′ . It follo ws that p is conta ined in A ′ . Thus B r ( y ) ∩ A ⊆ B r ( y ) ∩ A ′ . The other inclusion follo ws b y symmetry . ✷ W e call an aﬃne w all M ⊆ X thick if M is the intersectio n of three aﬃne apartments. W e ha v e the follo wing lo cal th ic kn ess criterion for wal ls. 4.17 Lemma Let A ⊆ X b e an aﬃne apartment, let p ∈ A and let M ⊆ A b e a w all cont aining p . Let r ∈ X p b e a panel in the wall determined b y M in X p . If r is conta ined in three d istin ct cham b ers of X p , then M is thic k. Pr o of. Let A p denote the apartment induced b y A in X p , let s ∈ A p b e the panel opp osite r and let a, b ∈ A p b e the t w o cham b ers conta ining s . W e repr esen t a, b, r, s by p -based W eyl simplices in A . By assumption, there is a cham b er con taining r whic h is opp osite b oth to a and to b . By 3.8 we can ﬁnd a p -b ased W eyl c ham b er c representing this c h am b er, such that ∂ c is adjacen t to ∂ A . Since the panel ∂ c ∩ ∂ A has a uniqu e p -based represen tativ e, this representa tiv e is contai ned in M . It follo ws that c ∩ A ⊆ M . Let A ′ denote the aﬃne apartment spann ed by c and a ; see [35, 1.12]. 25 p M a c A A ′ Then A ∩ A ′ is a half space wh ose b oundary is M and therefore M is th ic k. ✷ 4.18 Lemma Let p b e a p oint of an aﬃne apartmen t A ⊆ X . Th en p is thic k if and only if every w all of A con taining p is thic k. Pr o of. If p is thic k, then eve ry wall thr ough p is thick b y 4.17. Conv ersely , if ev ery wa ll of A con taining p is thic k, then all panels in A p are con tained in at least 3 c ham b ers. T h us, p is thick by 3.6. ✷ 4.19 Lemma Let A b e an aﬃne apartmen t and assum e that L, M ⊆ A are non parallel thick walls. Then th e reﬂection of L along M in A is also thic k. Pr o of. Let A ± ⊆ A denote the half spaces b ound ed by M and let A 0 ⊆ X b e a thir d half space with A ∩ A 0 = M . T here is a uniqu e aﬃne wa ll L ′ in the aﬃne apartment A ′ = A + ∪ A 0 extending L ∩ A + . By 4.17, L ′ is thic k. Now let L ′′ b e the wa ll in A 0 ∪ A − whic h extends L ′ ∩ A 0 . Again by 4.17, L ′′ is thic k. Finally , let L ′′′ b e the w all in A w hic h extends L ′′ ∩ A − . A third application of 4.17 yields that L ′′′ is thic k. M A − A + A 0 But L ′′′ is precisely the reﬂection of L along M . ✷ 4.20 Lemma The Euclidean bu ild ing X con tains a th ick p oin t if and only if ∂ A X is thick. Pr o of. By 3.8 thic kness of X p implies thic kn ess of ∂ A X . No w supp ose that ∂ A X is thic k, let A b e an aﬃne apartment, let c b e a W eyl cham b er of A and let M 1 , ..., M n b e th e wall s of A b ounding c . Since ∂ A X is thick, we can choose thic k w alls M ′ 1 , . . . , M ′ n in A s uc h that M ′ i is p arallel to M i for eac h i . The intersect ion of the thick walls M ′ 1 , . . . , M ′ n con tains a p oin t p . By 4.18 and 4.19, p is thic k. ✷ 26 The follo wing tric hotom y is analogous to the case of trees in 2.6. In higher dimension, h o wev er, we need no assumptions on group actions. 4.21 Prop osition S upp ose that X is a Euclidean building of dimension n ≥ 2 and that ∂ A X is irreducible and thic k. Let th ( X ) ⊆ X denote th e set of thic k p oin ts. There are the follo wing three p ossibilities. (I) There is a un ique thic k p oin t w hic h is contai ned in ev ery aﬃne apartment of X . (I I) T h e set of thic k p oin ts is a closed, discrete and cob ounded sub set in X and in ev ery apartmen t of X . (I I I) The set of thick p oin ts is dense in X an d in every apartmen t of X . Pr o of. W e noted in 4.20 that th ( X ) 6 = ∅ . F or an aﬃne apartment A ⊆ X we d enote b y R ( A ) ⊆ Isom( A ) the group generated b y reﬂections along the thic k walls in A . If p is a th ic k p oin t, then the R ( A )-stabilizer of p is R ( A ) p ∼ = W . W e start w ith tw o observ ations. (i) Supp ose that p is a thick p oin t in an aﬃne apartmen t A , and that M ⊆ A is a thic k wa ll not conta ining p . Suc h an apartment A exists if X con tains at least t wo thic k p oin ts. S ince the W eyl group W is irreducible and d im( A ) ≥ 2, the p oin t p is the in tersection of thic k w alls in A wh ic h are not parallel to M . It follo ws from 4.19 that the r eﬂection of p along M in A is again a thic k p oin t, so the R ( A )-orbit of p consists of thick p oin ts. (ii) F or an y t wo aﬃne apartments A, A ′ , there is a sequence of aﬃne apartments A = A 0 , . . . , A k = A ′ suc h that A j ∩ A j +1 is a half space. (Using galleries, this is easily seen to b e true for apartments in the spherical building at inﬁnity .) It is clear th at the s equence of w alls determined by these half spaces is thic k. Supp ose ﬁ rst that th ( X ) = { p } consists of a single p oint and that A is an aﬃne apartment con taining p . Then (i) shows that the thic k walls in A are precisely the ones p assing throu gh p . If A ′ is any other aﬃne ap artmen t, then (ii) and a simp le induction on k show that all apartment s A 0 , . . . , A k in the s equence connecting A and A ′ con tain p . This is case (I). Supp ose no w that th ( X ) conta ins at least t wo p oin ts p, q and that A is an apartmen t con taining p and q . Th en the R ( A )-orbit of q consists o f thic k p oin ts and is cob oun ded in A , b ecause the W eyl group W is ir reducible, an d b ecause R ( A ) conta ins translations. Let T ⊆ R ( A ) d enote the translational part (i.e. the k ernel of the action of R ( A ) on ∂ A ). An y closed subgroup of the v ector group ( R n , +) is a pro du ct of a free abelian group of ﬁnite rank and a v ector group. If T is discrete, then R ( A ) is an aﬃne reﬂection group. If T is not d iscrete, then the closure of T in I som( A ) consists of all translations in A (b ecause W acts irr educibly on the set of all translatio ns), so th ( X ) ∩ A is dense. If A ′ is an y other aﬃne apartmen t ha ving a half space in common with A , then the thic k wa lls in A propagate to thic k wa lls in A ′ . The isometry A ✲ A ′ also preserv es thic k p oints. W e obtain a canonical isomorphism R ( A ) ∼ = R ( A ′ ). F rom (ii) w e conclude that the isometry groups R ( A ) and R ( A ′′ ) are isomorphic f or any aﬃne apartmen t A ′′ ⊆ X , and that this isomorphism maps th ( X ) ∩ A on to th ( X ) ∩ A ′′ . In the discrete case, we h a ve (I I), whereas the nond iscrete case is (I I I). ✷ 4.22 C orollary Let X b e as in 4.21 (I). Th en X ∼ = EB( ∂ A X ) is a E uclidean cone. Pr o of. Let p ∈ X b e the unique thic k p oin t and let A b e an aﬃne apartmen t. Since we are in ca se (I), the p oin t p is con tained in A . Th e canonical surjectiv e 1-Lipsc h itz m ap EB( ∂ A X ) ✲ X send s therefore EB( ∂ A ) isometrically onto A . Sin ce an y t wo points in EB( ∂ A X ) are in some apartment , EB( ∂ A X ) ✲ X is an isometry . ✷ 27 The follo wing observ ation will n ot b e used, b ut it illustrates ho w simp licial aﬃne buildings ﬁt in to th e picture. In the r emaining case (I I I), X is a nond iscrete Euclidean b uilding. 4.23 C orollary Let X b e as in 4.21 (I I). Th en X is the metric realization of an aﬃne simp licial building. Pr o of. T he group s R ( A ) are aﬃne Co xeter group s; s ee [8 , Ch.VI]. In this w a y , ev ery apartment has a canonical simplicial structure as a Co xeter complex. The axioms of an aﬃne simplicial bu ilding follo w from 4.2. ✷ In general, a Euclidean build ing is not d etermined b y its sph erical b uilding at in ﬁnit y . F or example, ∂ A EB( ∂ A X ) = ∂ A X. Some additional data are needed, wh ic h are enco ded in the panel trees. The follo wing material is du e to Tits [46]. A pr o of of 4.28 w ith all details ﬁ lled in can b e f ound in [20]. 4.24 W all t rees and panel tree s Let ( X, A ) b e a Eu clidean buildin g and let a, b b e a p air of opp osite panels in the sp herical b uilding at inﬁnit y . L et X ( a, b ) d enote the union o ver all aﬃne apartmen ts in A wh ose b oundary conta ins a and b , X ( a, b ) = [ { A | A ⊆ X aﬃne apartmen t in A and a, b ∈ ∂ A } . Then X ( a, b ) is a Euclidean building, w hic h factors metrically as X ( a, b ) = T × R n − 1 , where T is a leaﬂess tree; see [25, 4.8 .1], [40, 3.9] [46]. W e call this tree T the wal l tr e e asso ciated to ( a, b ), b ecause it dep end s only on the unique w all of ∂ A X co nta ining a and b . In th e notation of 3.15, the spher ical building at inﬁ nit y of X ( a, b ) is ∂ ( X ( a, b )) = ( ∂ A X )( a, b ) . If X is metrically complete, then T and X ( a, b ) are also m etrically complete [44]. 4.25 Lemma Let X b e a Euclidean b uilding and let a, b, b ′ b e panels in ∂ A X . Sup p ose that b and b ′ are opp osite a . Then ther e is a uniqu e isometry X ( a, b ) ✲ X ( a, b ′ ) wh ic h ﬁxes X ( a, b ) ∩ X ( a, b ′ ) p oint wise. Pr o of. Let A b e an a ﬃne apartment whose b oundary con tains a, b , and let A ′ b e the corresp ondin g aﬃne apartment wh ose b oundary con tains Res( a ) ∩ ∂ A and b ′ ; see 3.15. S ince A ∩ A ′ con tains W eyl c h am b ers, there is a uniqu e isometry A ✲ A ′ ﬁxing A ∩ A ′ . This pro ve s the u niqueness of the isometry . F or the existence, w e show that these isometries A ✲ A ′ of the individu al aﬃne apartmen ts ﬁt together. Let A 1 , A 2 ⊆ X ( a, b ) b e t w o aﬃne apartment s con taining a p oint x ∈ A 1 ∩ A 2 . Corr esp ondingly , w e hav e c ham b ers c i , d i ∈ Res( a ) suc h that c i , pro j b d i spans ∂ A i ; see 3.15. If t wo of these four c hambers coincide, sa y c 1 = c 2 , then A 1 and A 2 ha v e the x -based W eyl c h am b er ˜ c represen ting c 1 in common. Then A 1 , A 2 , A ′ 1 and A ′ 2 ha v e a su b-W eyl c hamb er of ˜ c in common, and th erefore the t w o isometries A 1 ✲ A ′ 1 and A 2 ✲ A ′ 2 map x to the same p oin t x ′ . 28 If c 1 , c 2 , d 1 , d 2 are pairwise diﬀeren t, then x is also in the aﬃne apartment determined b y c 1 , d 2 or c 1 , c 2 , b ecause X ( a, b ) is a p ro duct of a tree and Eu clidean sp ace. The pr evious argument, applied t w ice, sho ws that the v arious apartmen t isometries coincide on x . T h us w e ha v e a well -deﬁned bijectio n whic h is apartment-wise an isometry . Sin ce any t wo p oints are in some aﬃn e apartment, th e map is an isometry . ✷ 4.26 If b ′ v aries o ver the pan els opp osite a , we obtai n a family of tr ees whic h are pairwise canonically isomorphic. T his canonical isomorph ism t yp e of a tree is the p anel tr e e T a asso ciated to a . If a 0 a 1 · · · a k is a path in the opp osition graph, then th e isomorphisms ( ∂ A X )( a 0 , a 1 ) ∼ = ( ∂ A X )( a 1 , a 2 ) ∼ = · · · from 3.15 are accompanied b y isometries X ( a 0 , a 1 ) ∼ = X ( a 1 , a 2 ) ∼ = · · · . In particular, th e group of pr o jectiviti es Π ∂ A X ( a ) acts on the wall tree and on th e panel tree T a . If we r estrict to ev en p ro jectivities, the action on th e Euclidean factor is trivial. If the b uilding is th ic k and the typ e of the panel a is not isolated, then the action of Π + ∂ A X 1 ( a ) on the ends of T a is 2-transitiv e by 3.14 The branch p oints in the panel tr ees corresp ond to th e thic k walls in X . Assumin g that ∂ A X is irreducible, w e ha v e the follo wing consequence of 4.21. If one p anel tree T a is of t yp e (I), then ev ery panel tree is of t yp e (I) and X is a Euclidean cone ov er | ∂ A X | as in 4.3. If one p anel tree is of t yp e (I I), then ev ery panel tree is of type (I I) and X is simp licial. Th e remaining p ossibilit y is that ev ery panel tr ee is of type (I I I). 4.27 Ecological b oundary isomorphisms L et X 1 , X 2 b e irreducible Euclidean b uildings. Assume that ∂ A 1 X 1 and ∂ A 2 X 2 are thick and that ϕ : ∂ A 1 X 1 ✲ ∂ A 2 X 2 is an isomorphism. Assum e moreo ver that for eve ry panel a ∈ ∂ A 1 X 1 , there is a tree isometry ϕ a : T a ✲ T ϕ ( a ) . If for eac h p anel a , th e map ∂ ϕ a : ∂ T a ✲ ∂ T ϕ ( a ) coincides with the restriction of ϕ to Res( a ), th en ϕ is called tr e e- pr eserving or e c olo gic al . The follo wing is [46, Th m. 2], see also [20, Ch . 7]. 4.28 The orem (Tits) Let ( X 1 , A 1 ) and ( X 2 , A 2 ) b e Euclidean b uildings. Assu me th at ∂ A 1 X 1 is thic k and irredu cible. If ϕ : ∂ A 1 X 1 ∼ = ✲ ∂ A 2 X 2 is an ec ological isomorphism, then ϕ extend s to an isomorphism Φ : ( X 1 , A 1 ) ∼ = ✲ ( X 2 , A 2 ) . ✷ The irr educibilit y is actually not imp ortan t for the p ro of, but the result is stated in this wa y in [46]. F or our p urp oses, the irr educible v ersion suﬃces. W e r emark that the follo wing interesting problem is op en. 4.29 Conjecture L et X 1 , X 2 b e Euclide an bu ildings of typ e (II) and dimension n ≥ 2 . A ssume that ∂ cpl X 1 is i rr e ducible. Then every isomorp hism ∂ cpl X 1 ✲ ∂ cpl X 2 extends uniquely to an isometry X 1 ✲ X 2 . 29 This is kno w n to b e true if ∂ cpl X 1 is Moufang [49, 27.6]. The pro of is algebraic and uses the fact that a ﬁeld admits at most one discr ete complete v aluation. The conjecture is false for Euclidean bu ildings of t y p e (I I I), since for example C adm its inﬁnitely many nonisomorphic nondiscrete complete v aluations, corresp onding to nonisomorph ic Euclidean buildings of type e A m . The b ound ary is alw a ys the spherical building of SL m +1 C . 5 Coarse equiv alences of Euclidean bu ildings In this ﬁnal section w e pro v e, among other things, Theorem I I and I I I fr om the introdu ction. An imp ortant tec hnical to ol is the ’higher dimensional Morse Lemma’ 5.9. The main step in this result, in turn, is the follo wing top olog ical rigidit y theorem [25, 6.4.2] [28, Sec. 4]. 5.1 Theorem (T opological rigidit y of a ﬃne apart men t s) Let f : X ✲ Y b e a homeomorphism of metricall y complete Euclidean buildings. If A ⊆ X is an aﬃne apartmen t, then f ( A ) is an aﬃne apartment in the complete apartment system of Y . Sketch of pr o of. W e giv e a n o utline of the shea f-theoretic proof from [28, Sec. 6]. O n any Hausd orﬀ space X , there are the follo win g t wo pr eshea ves. Th e lo c al ho molo gy pr eshe af assigns to every op en subset U ⊆ X the singular homology group H ∗ ( X, X − U ). The stalk of the corresp ond ing sh eaf H ∗ ( X ) at a p oin t p ∈ X is the lo cal homology group H ∗ ( X ) p ∼ = H ∗ ( X, X − { p } ) . The sec ond presheaf assigns to U the p oset C l s ( U ) = { A ∩ U | A ⊆ X is closed } of all rela tiv ely closed subsets of U . Its stalk Cls p consists of germs of closed sets at p . There is a natural transformation supp b et ween th ese (pre)shea v es wh ic h assigns to ev ery r elativ e cycle c ∈ H ∗ ( X, X − U ) its supp ort supp ( c ) = { p ∈ U | c has nontrivial im age in H ∗ ( X, X − { p } ) } [25, 6.2] [28, Sec. 4]. If X is a Eu clidean buildin g, then the r esidue X p is by deﬁ nition a s ub-p oset of Cls p . The main step in the pro of of 5.1 is to sho w that this sub -p oset can b e d escrib ed by means of the transformation supp . The reason is as follo ws. F or any CA T(0) space X , the natural map X − { p } ✲ Σ p X whic h assigns to a p oin t q 6 = p the direction of the geo desic segmen t [ q , p ] in the (completed) sp ac e of dir e ctions Σ p X is a homotop y equiv alence [28, Th m. A]. In p articular, there is a natural isomorph ism in homology H ∗ +1 ( X, X − { p } ) ∼ = ✲ ˜ H ∗ (Σ p X ) . If X is a Euclidean buildin g, then the space of directions Σ p X is th e geometric realization of th e residue X p . Th e reduced homology (of the geometric realizatio n) of the sph erical buildin g X p is th en a free Z -mo d ule spanned by all apartments cont aining a given c ham b er, see [28, Sec. 5], the Steinb e r g mo dule (this is essen tially th e con tents of the Solomo n-Tits Theorem). I n particular, the dimension n = dim( X ) can b e read oﬀ from th e lo cal homology groups (alternativ ely , the n u m b er n coincides 30 with the co v ering dimension of X [28, 7.1] [30, 3.3] and hence is a top olog ical in v arian t). Th is allo w s us to describ e the image of the transformation H ∗ ( X ) p supp ✲ Cls , p . If the residue X p is thic k, then the simplices of X p are precisely the indecomp osable elements in the distribu tiv e p oset lattice generated b y supp ( H ∗ ( X ) p ), see [28, Cor. 6.6] (we call an elemen t indecomp osable if it is not a union of ﬁnitely man y strictly smaller elemen ts). This is clearly a top ologica l inv arian t of the space X . In general, we ha ve b y 3.5 a thic k redu ction X p ∼ = S k ∗ | B 0 | , with a thic k spherical building B 0 . In this case the indecomp osable elemen ts in the image of supp corresp ond to the simplices of the thic k part B 0 , s ee [28, 5.6]. In particular, w e can read oﬀ the n umber k f rom the diﬀerence b et we en the dimension of the simplicial complex B 0 and the degree j for whic h ˜ H j ( X − { p } ) 6 = 0. It follo w s th at the homeomorphism map s a small op en ball B r ( p ) ∩ A in the aﬃn e apartmen t A ⊆ X to a small op en set in some aﬃne apartmen t in Y . Thus f ( A ) ⊆ Y is a complete simply connected ﬂat Riemann ian manifold, and therefore isometric to R n . By 4.13, f ( A ) is an aﬃne apartment in the complete apartmen t system of Y . See [28, S ec. 6] for deta ils and a more general result. If ∂ A X is thic k and irreducible and if n ≥ 2, then the completeness assum ption can b e drop p ed. ✷ W e also hav e the follo wing coroll ary of the p ro of. 5.2 Corollary Assume th at X and Y are metrically complete Euclidean buildings, and that ∂ cpl X and ∂ cpl Y are thick. If f : X × R m 1 ✲ Y × R m 2 is a homeomorphism, th en m 1 = m 2 . Pr o of. Let p ∈ X b e a p oin t. Th en the r esidue at p × v ∈ X × R m 1 decomp oses as ( X × R m 1 ) p × v = | B 0 | ∗ S k , where B 0 is a thic k spherical b uilding and k ≥ m 1 . If p is thic k, then k = m 1 . The same applies to Y , hence m 1 = m 2 . ✷ Theorem 5.1 is th e ‘hard part’ of the pro of of T heorem 5.9 b elo w. The t ransition from h omeomorphism rigidit y to coarse rigidit y , on the other hand, is mainly a m atter of elemen tary logic. See also [47] and [29]. 5.3 The language of Euclidean buildings Recall fr om 1.7 our language of metric space L ms = { + , · , ≤ , 0 , 1 , d, X , R } . W e no w enlarge this language so that w e can s tate resu lts ab out Eu clidean b uildings. W e ﬁ x a spherical Co xeter group W = h I | ( ij ) m ij = 1 i and add constan ts i ∈ I for its generators and the m i,j to the language. W e also add a unary predicate W for the W eyl group. This allo ws u s to include the spherical W eyl group W as well as the aﬃn e W eyl group W R n in to our structure. Then we add function symbols for the coro ots α i : R n ✲ R , w hose k ernels are the reﬂection h yp er planes corresp onding to th e generators i ∈ I . Th is allo ws us to describ e W eyl c ham b ers, half spaces, w alls and W eyl simplices in R n in our la nguage. The last i ngredient t hat is missing are t he coord inate c harts ϕ : R n ✲ X . The n u m b er of c harts will dep end very m uch on the Euclidean b uilding X . In order to allo w some ﬂexibilit y , w e view the charts as a family of maps { ϕ ℓ : R n ✲ X } ℓ ∈ L . 31 Th us we add a unary predicate L for the ind ices ℓ that lab el the c harts, and one sin gle n + 1-ary function symb ol ϕ : L × R n ✲ X , whose ﬁrst en try is the lab el ℓ ∈ L . This is the language L eb of Euclidean buildings. It is straigh tforw ard to see that a Euclidean building ( X , A ) can b e viewed as an L eb -structure. The axioms (EB1 )–(EB5) ca n b e stated without d iﬃculties in this language , and so can b e most of the r esults ab out Euclidean buildin gs that we p ro v ed in Section 4. Belo w we add a few more elements to the language L eb , for example the v ector distance function, or a second Eu clidean buildin g Y and a coa rse map f : X ✲ Y b etw een the t wo . W e note also that the axioms (EB5’) and (EB5 + ) can b e stated if we add one fu nction symb ol ρ for the retractio ns, ρ ℓ : X ✲ ϕ ℓ ( R n ) ha ving as its ﬁrst argument the index ℓ d escribing the target apartmen t A = ϕ ℓ ( R n ). The cen ter of the retraction is ϕ ℓ (0). 5.4 Ultrapro ducts of E uclidean buildings Supp ose n o w th at K is a counta bly in ﬁnite set and that D is a n onprincipal ultraﬁlter on K . S upp ose that ( X k , A k ) k ∈ K is a family of Euclidean buildings, all of which are mo deled on the same sp herical Coxe ter group W . W e ma y view them as L eb -structures X k and tak e their ultrapro d uct X D . W e n ote that W D = W by 1.11. By Los’ Theorem 1.10, the ultrapro d uct satisﬁes the axioms (EB1)–(EB5 ), except that the real num b ers are no w rep laced by the real closed ﬁeld ∗ R and th at the apartments are mo d eled on ∗ R n . The co ord inate c h anges are n o w describ ed by the nonstand ard aﬃne W eyl group W ⋉ ∗ R n . One can sho w that X D is a gener alize d aﬃne bui lding in the sense of Bennett [3 ] [20] [43]. W e call X D a nonstandar d Euc lide an building , with nonstandar d charts , nonstandar d ap artments , nonstandar d Weyl simplic es , and so on. 5.5 Ultralimits of Euclidean buildings Sup p ose that X D is an ultrapro duct of of a family of Euclidean bu ildings ( X k , A k ) as in 5.3 an d 5.4. Let p ∈ X D and let r > 0 b e a nonstandard real. T hen w e ha ve the ultralimit C ( X D , p, r ) of th e un derlying metric spaces X k as in 1.13 . Recall that X ( p,r ) D = { x ∈ X D | 1 r d ( p, x ) ∈ ∗ R ﬁn } and put ( ∗ R n ﬁn ) (0 ,r ) = { v ∈ ∗ R n | 1 r || v || ∈ ∗ R ﬁn } = { v r | v ∈ ∗ R n ﬁn } . Supp ose that ϕ ℓ is a nonstandard c hart with ϕ ℓ (0) ∈ X ( p,r ) D . Then we hav e a commuting diagram ( ∗ R n ﬁn ) (0 ,r ) ϕ ℓ ✲ X ( p,r ) D R n ❄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˜ ϕ ℓ ✲ C ( X D , p, r ) ❄ where the vertica l arro ws iden tify p oin ts at inﬁn itesimal 1 r d -distance. T he m ap ˜ ϕ ℓ is an isometric injection, and we let ˜ A denote the set of all maps ˜ ϕ ℓ that arise in th is wa y . The corresp ond ing aﬃn e apartmen ts (the images of th e ˜ ϕ ℓ ) will b e denoted by ˜ A ⊆ C ( X D , p, r ). The follo win g result is pro ve d in [25, 5.1.1]. A more general resu lt is p ro v ed by Sch wer and S truyv e in [43, 6.1]. W e remark that X ( p,r ) D is also a generalized aﬃn e bu ilding in the sense of Bennett [3]. 32 5.6 Theorem Let ( X k , A k ) k ∈ K b e a counta bly in ﬁ nite family of Euclidean bu ildings, with a ﬁxed spherical Co xeter group W . Let C ( X D , p, r ) and ˜ A b e as in 5.5. Then the ultralimit C ( X D , p, r ) is a metrically complete Euclidean bu ilding, w ith ˜ A as its complete set of ap artments. Pr o of. F rom the construction of ˜ A and Los’ T heorem it is clear that ˜ A satisﬁes the axioms (EB1) and (EB2). Th e retractions X D ✲ A on to the nonstandard aﬃne apartments are 1-Lipsc hitz and descend therefore to retracti ons C ( X D , p, r ) ✲ ˜ A , hen ce axiom (EB5 + ) h olds. The r emaining tw o axioms require more work. Axiom (EB3) holds. W e note that C ( X D , p, r ) is CA T(0) by 1.1 5. Since the aﬃne apartmen ts are con vex, their intersec tions are also con v ex. No w we add the vec tor distance Θ to the structure. It is clear that this giv es us a n onstandard ve ctor distance on X ( p,r ) D ⊆ X D . Su pp ose that x, x ′ , y ∈ X ( p,r ) D and that x and x ′ ha v e inﬁnitesimal 1 r d -distance. Th en Θ( y , x ) and Θ( y , x ′ ) also hav e inﬁnitesimal 1 r d -distance. F rom the symmetry of the vect or distance 4.6, we conclude that Θ( x, y ) and Θ( x ′ , y ) ha v e inﬁnitesimal distance. T his implies that the nonstandard v ector distance descends to an ordinary v ector distance ˜ Θ : C ( X D , p, r ) × C ( X D , p, r ) ✲ a 0 . It n o w follo ws f rom 4.7 that co ordinate changes b et w een charts in C ( X D , p, r ) are given by elemen ts of W R n , h ence (EB3) holds. Axiom (EB4) holds. W e ﬁrst show t w o auxiliary results. Claim 1. L et c, c ′ ⊆ X D b e nonsta ndar d Weyl chamb ers with ∂ c = ∂ c ′ . If the tips z an d z ′ of c and c ′ ar e in X ( p,r ) D , then ther e exists a nonstanda r d Weyl c hamb er a ⊆ c ∩ c ′ whose tip w is also in X ( p,r ) D . Let [ z , ζ c ) b e the nonstandard central ray of c (we may add these r a ys to our structure and language). This ra y int ersects th e W eyl cham b er c ′ and there is a unique p oin t 2 w w ith [ z , ζ c ) ∩ c ′ = [ w, ζ c ). Let s = 1 r d ( w, z ) ∈ ∗ R . W e claim that s is a ﬁnite n onstandard real. S upp ose to the con trary that s is in ﬁnite. L et [ x, ζ c ) ⊆ c ′ b e the maximal ra y in c ′ extending [ w, ζ c ). Th en the three p oin ts z , z ′ , x h a ve mutually inﬁn itesimal 1 r s d -distance, while w an d z hav e 1 r s d -distance 1 (mo d ∗ R inf ). It follo w s that the three nonstandard Alexandro v angles at w b etw een the segmen ts [ w , z ] , [ w, z ′ ] , [ w , x ] are inﬁ nitesimally small (w e ma y also ad d the Alexandro v angles to our structure and language , so they are d eﬁned in X D ). w z z ′ x This contradict s 4.15. Thus s is ﬁnite and hence w ∈ X ( p,r ) D . No w let a b e the unique nonstandard w -b ased W eyl cham b er with ∂ a = ∂ c . Th us Claim 1 is prov ed. W e note that ev ery W eyl cham b er ˜ c ⊆ C ( X D , p, r ) arises f rom some nonstandard W eyl c ham b er c whose tip is in X ( p,r ) D . Supp ose that ˜ c, ˜ c ′ ⊆ C ( X D , p, r ) are W eyl cham b ers . W e choose corresp onding nonstandard W eyl c hambers c, c ′ ⊆ X D in suc h a w a y that ∂ c, ∂ c ′ ha v e minimal ga llery distance m ∈ N in the sph erical buildin g at inﬁ nit y of X D . 2 The p oint ex ists by Los’ Theorem b ecause it is deﬁnable in the stru cture, hence it do es not matter that the ordered ﬁeld ∗ R is n on-arc himedean. 33 Claim 2. In this situation, ther e is an nonstanda r d aﬃne ap artment A c ontaining a p oint w ∈ X ( p,r ) D and a g al lery of w -b ase d nonstanda r d Weyl chamb ers c 0 , . . . , c m , with ∂ c = ∂ c 0 and ∂ c ′ = ∂ c m . W e pr o ceed b y ind uction on m , the case m = 0 b eing trivial. Assu me no w that c 0 , . . . , c m − 1 are w -b ased W eyl c ham b ers con tained in a nonstandard aﬃne apartmen t A , with w ∈ X ( p,r ) D . Let a b e the u nique w -based W eyl c hamb er w ith ∂ a = ∂ c ′ . If a is con tained in A we are done, with c m = a . If a is not con tained in A , th en we let M ⊆ A denote the nonstandard wall that separates a and c m − 1 . Let H ⊆ A denote the closed h alf space b ounded by M whose b oun dary at inﬁ nit y con tains ∂ c m − 1 . If H and X ( p,r ) D ha v e a p oint w ′ in common, th en we rep lace c 0 , . . . , c m − 1 b y their w ′ -based translates in A , and we let c m denote the unique w ′ -based W eyl cham b er w ith ∂ c m = ∂ c ′ . w ′ M H A c m − 1 a Let ¯ a b e the u nique w ′ -based W eyl cham b er in H whose germ near w ′ is opp osite to the germ of a . Then a, ¯ a are con tained in a unique nonstandard aﬃne apartmen t A ′ con taining c 0 ∪ · · · ∪ c m , hence w e are done. w ′ M H A ¯ a a A ′ Supp ose that H ∩ X ( p,r ) D = ∅ . Th en M separates w from ∂ c m − 1 , hence a has th e same germ near w as c m − 1 . a M H A c m − 1 This situation is excluded b y our assu mptions. Thus we ha v e pr o ved C laim 2. In the setting of Claim 2, it follo ws fr om Claim 1 that ˜ c ∩ ˜ c 0 con tains a W eyl c hamber, and that ˜ c ′ ∩ ˜ c m con tains a W eyl c h am b er. Therefore (EB4) holds. The atlas ˜ A is maximal. W e h a ve to show that every geo desic lin e and ra y is contai ned in some aﬃne apartmen t [35, 2.18]. Let γ : R ✲ C ( X D , p, r ) b e a geo desic line. W e c ho ose p oints x k ∈ X ( p,r ) D corresp onding to γ ( k ) ∈ C ( X D , p, r ), for k = 0 , ± 1 , ± 2 , . . . . F or k ≥ 0, let A k b e a nonstand ard aﬃne 34 apartmen t conta ining { x − k , x k } . Then x ℓ is 1 r d -inﬁnitesimally close to A k for all ℓ = − k , . . . , k . Hence w e hav e th e follo win g pr op ert y of the set T = { x k | k ∈ Z } . F or ev ery ﬁnite subset S ⊆ T there exists a non standard aﬃne apartment A S suc h that all elemen ts of S are 1 r d -inﬁnitesimally close to A S . By 1.12 there exists a nonstand ard aﬃne ap artmen t A T suc h that all memb ers of T are 1 r d -inﬁnitesimally close to A T . Th us γ ( k ) ∈ ˜ A T holds for all k ∈ Z . Since ˜ A T is con v ex, we ha v e γ ( R ) ⊆ ˜ A T . The reasoning for ra ys is completely analo gous. ✷ The follo win g consequence is often useful. Note that the metric completion of a n on-complete Eu- clidean building need n ot b e a Eu clidean b uilding [28, 6.9], see also [32]. 5.7 Corollary Let ( X, A ) b e a Eu clidean buildin g. Then there exists a complete Euclidean building ( ¯ X , ¯ A ) of the same type, with X ⊆ ¯ X and A ⊆ ¯ A . The action of the automorphism group of ( X, A ) extends to an action on ( ¯ X , ¯ A ) . Pr o of. Let X D b e the ultrap ow er of X and pu t ¯ X = C ( X D , p, r ). The d iagonal embedd ing X ✲ X D extends to an embed ding X ✲ ¯ X . The automorphism group of ( X, A ) acts diagonally on X D , on X ( p,r ) D , and hence on ¯ X in a natural wa y . ✷ 5.8 The structure C S upp ose that f : X ✲ Y is a coarse equiv alence of Euclidean bu ildings. W e consider the structure C consisting of the t wo Eu clidean buildings and the map f , and w e tak e the ultrap ow er C D of this str ucture. If p ∈ X D is any b asep oin t and if r ∈ ∗ R is inﬁn itely large, then f indu ces a bi-Lipschitz homeomorphism s C ( f ) : C ( X D , p, r ) ✲ C ( Y D , f ( p ) , r ) b y 1.16. By 5.6, the asymptotic cones C ( X D , p, r ) an d C ( Y D , f ( p ) , r ) are metrically complete Euclidean buildings with complete apartment systems, and C ( f ) maps aﬃn e ap artmen ts to aﬃne apartmen ts b y 5.1. W e also n ote the follo wing. T he ordered ﬁeld ∗ R is non-arc himedean, so a b ou nded set of nonstandard reals will in general n ot ha v e a sup rem um. Nev ertheless, the quan tity H d B r ( y ) ( f ( A ) , A ′ ) = H d ( f ( A ) ∩ B r ( y ) , A ′ ∩ B r ( y )) is deﬁned for all nonstandard aﬃne apartments A ⊆ X D , A ′ ⊆ Y D , p oints y ∈ Y D and p ositive nonstandard reals r > 0. The reaso n for this is that w e can add H d B r ( y ) ( f ( A ) , A ′ ) as an R ∪ {∞} - v alued function (dep ending on argumen ts ℓ, ℓ ′ , y , r , with A = ϕ ℓ ( R n ), A ′ = ϕ ′ ℓ ′ ( R n )) to the structure C and then take its u ltrap o w er. Los’ Theorem 1.10 guaran tees th at this function has exactly th e in tended meaning in C D , namely that of an ∗ R -v alued Hausdorﬀ distance 3 . These ob serv ations are the main steps in our pr o of of the follo wing theorem [25, 7.1.5]. 5.9 Theorem (Higher dimensional Morse Lemma) Let f : X ✲ Y b e a co arse equiv alence b etw een E uclidean bu ildings. There exists a real constan t r f > 0 suc h that the follo w in g holds. F or ev ery aﬃne apartmen t A ⊆ X th er e is a unique aﬃne apartmen t A ′ in the complete apartmen t system of Y w ith H d ( f ( A ) , A ′ ) ≤ r f . 3 The p oint is th at we measure only the Hausdorﬀ distance b etw een so-called internal or deﬁnable sets. 35 W e ﬁ rst prov e a weak er statemen t in the ultrapro duct. 5.10 Lemma Let C b e as in 5.8. L et A ⊆ X D b e a n onstandard aﬃne apartmen t and let y ∈ f ( A ) . Let r ≥ 1 b e a nonstand ard real. There exist a nonstandard aﬃn e apartmen t A ′ ⊆ Y D and a ﬁnite nonstandard real num b er s ≥ 0 suc h th at H d B εr ( y ) ( f D ( A ) , A ′ ) ≤ s holds for all nonstandard ε with 1 2 ≤ ε ≤ 1 . Pr o of. Th is is clear if r is ﬁnite: we m a y choose any nonstandard aﬃne apartmen t A ′ con taining y an d c ho ose an y real n u m b er s ≥ 2 r . S o su pp ose that r is inﬁn ite. W e c h o ose a p reimage x ∈ A of y . Sin ce r is inﬁ nite, we hav e b y L emma 1.16 a b i-Lipsc hitz map C ( f ) : C ( X D , x, r ) ✲ C ( Y D , y , r ) . Let ˜ A ⊆ C ( X D , x, r ) denote the aﬃne corresp onding to A . By Theorem 5.1 C ( f ) maps ˜ A on to an aﬃne apartmen t ˜ A ′ in th e complete apartment system of the Euclidean buildin g C ( Y D , y , r ). By 5.6 there is a n onstandard aﬃne apartmen t A ′ ⊆ Y D corresp onding to ˜ A ′ . Supp ose no w that ε is a nonstandard real with 1 2 ≤ ε ≤ 1. Let s ε = H d B εr ( y ) ( f ( A ) , A ′ ) . Since w e ha ve in C ( Y D , y , r ) that B ε ( ˜ y ) ∩ C ( f )( ˜ A ) = B ε ( ˜ y ) ∩ ˜ A ′ , the quotien t s ε /r is inﬁnitesimally small. W e claim that s ε is ﬁnite. Su pp ose to the co ntrary that s ε > 0 is in ﬁnite. T here is either a p oin t z ∈ A ′ ∩ B εr ( y ) such that B s ε / 2 ( z ) ∩ f ( A ) = ∅ or a p oin t z ∈ f ( A ) ∩ B εr ( y ) such that B s ε / 2 ( z ) ∩ A ′ = ∅ . Because s ε is in ﬁnite, we ha v e a b i-Lipsc h itz map C ( f ) : C ( X D , w, s ε ) ✲ C ( Y D , z , s ε ) (for a suitably chosen p oin t w ∈ X D ). In C ( Y D , z , s ε ), the sets C ( f )( ˜ A ) and ˜ A ′ are aﬃne apartmen ts, and C ( f )( ˜ A ) 6 = ˜ A ′ b y the choi ce of z . It is also clear from the constru ction that C ( f )( ˜ A ) and ˜ A ′ ha v e Hausdorﬀ distance at most 1 in C ( Y D , z , s ε ). This is imp ossible by [25, 4.6.4] [35, p. 10], or by 4.16. Th us s ε has to b e ﬁnite. No w w e claim th at the set { s ε | 1 2 ≤ ε ≤ 1 } ⊆ ∗ R has a ﬁnite upp er b ound. Supp ose that th is is false. Then w e ﬁnd f or ev ery k ∈ N an ε k with 1 2 ≤ ε k ≤ 1 suc h that s ε k ≥ k . By 1.12 w e ﬁnd an ε with 1 2 ≤ ε ≤ 1 suc h that s ε is inﬁnite, a contradict ion. Hence { s ε | 1 2 ≤ ε ≤ 1 } ⊆ ∗ R has a ﬁnite upp er b ound s . ✷ The next lemma says that the num b ers s o ccurrin g in the previous lemma can b e b ounded uniformly from ab o v e. This is another application of 1.1 2. 5.11 Lemma Let C b e as in 5.8. Th ere exists a ﬁnite real co nstant s ≥ 0 suc h that the follo wing holds. F or ev ery r ≥ 1 , ev ery nonstandard aﬃne apartment A ⊆ X D and ev ery x ∈ A there exists a nonstandard aﬃne apartmen t A ′ ⊆ Y D suc h that H d B εr ( f ( x )) ( f ( A ) , A ′ ) ≤ s holds for all 1 2 ≤ ε ≤ 1 . Pr o of. Assume that this is false. Then we can ﬁnd for ev ery k ∈ N a p ositiv e nonstandard real r k ≥ 1, an ε k with 1 2 ≤ ε k ≤ 1, a nonstandard aﬃne apartment A k ⊆ X D , a p oin t x k ∈ A k suc h that 36 for ev ery n onstandard aﬃne apartment A ′ ⊆ Y D w e ha ve H d B ε k r k ( f ( x k )) ( f ( A k ) , A ′ ) ≥ k . By 1.12 th ere exist a n onstandard aﬃne apartment A ⊆ X D , a p oin t x ∈ A , a nonstandard r eal r ≥ 1 and an ε w ith 1 2 ≤ ε ≤ 1 s uc h that H d B εr ( f ( x )) ( f ( A ) , A ′ ) is inﬁnite for all nonstandard aﬃne apartmen ts A ′ ⊆ Y D . This con tradicts Lemm a 5.10. ✷ By Los’ Theorem 1.10 we h a ve the follo w ing immediate consequence for the co arse equ iv alence that w e started with. 5.12 C orollary Let f : X ✲ Y b e a a coarse equiv alence b et ween E uclidean buildin gs. There exists a real constant r f ≥ 0 s u c h that the follo wing holds. F or ev ery r ≥ 1 , ev ery aﬃne apartment A ⊆ X , ev ery x ∈ A there exists an aﬃne apartmen t A ′ ⊆ Y suc h that H d B εr ( f ( x )) ( f ( A ) , A ′ ) ≤ r f holds for all 1 2 ≤ ε ≤ 1 . ✷ Pr o of of the higher dimensional Morse L emma 5.9. Let A ⊆ X b e an aﬃne apartmen t, let x ∈ A and put Z = f ( A ) and z = f ( x ). Let r f b e as in 5.12. F or ev ery s ≥ 1 we ma y by 5.12 c ho ose an aﬃne apartmen t A s ⊆ Y su c h that H d B εs ( z ) ( Z, A s ) ≤ r f holds for all s ∈ [ 1 2 , 1]. Let c = c 2 r f b e the constant from 4.16, corresp onding to Hausdorﬀ distance 2 r f . F or s ≥ 1 we h a ve H d B s + c ( z ) ( Z, A s + c ) ≤ r f and H d B s + c ( z ) ( Z, A 2( s + c ) ) ≤ r f , w hence H d B s + c ( z ) ( A s + c , A 2( s + c ) ) ≤ 2 r f and, b y 4.16, B s ( z ) ∩ A s + c = B s ( z ) ∩ A 2( s + c ) . Let w = π A s + c ( z ) = π A 2( s + c ) ( z ). W e hav e then, for s > r f , B s − r f ( w ) ∩ A s + c = B s − r f ( w ) ∩ A 2( s + c ) . Then A r f + c , A 2( r f + c ) , A 4( r f + c ) , . . . , A 2 k ( r f + c ) , . . . give s us a nested sequence of metric op en n -balls (where n = dim ( X )) with centers w and of radii 2 k ( r f + c ) − c . Th eir u nion A ′ = ∞ [ k =0  B 2 k ( r f + c ) − c ( w ) ∩ A 2 k ( r f + c )  is isometric to R n and hence by 4.13 an aﬃn e apartmen t in the complete apartmen t system of Y . F rom the construction of A ′ it is clear that H d ( Z , A ′ ) ≤ r f . ✷ W e also hav e the follo wing. 37 5.13 Prop osition Let X, Y b e Euclidean buildin gs whose spherical buildings at in ﬁ nit y are thic k. Let f : X × R m 1 ✲ Y × R m 2 b e a coarse equiv alence. Th en m 1 = m 2 . Pr o of. W e c ho ose an inﬁnite p ositiv e nonstandard real r and a thic k p oin t p ∈ X , see 4.20. W e then consider the asymptotic cone C (( X × R m 1 ) D , r , p × 0) ∼ = C ( X D , r , p ) × R m 1 with resp ect to the constant f amilies X k = X and p k = p . I t is clear that the constant sequence ( p k ) k ∈ K represent s a thick p oint in the ultrapro du ct X D , and hence also in the asymptotic cone C ( X D , r , p ). In particular, the spherical bu ilding at inﬁn it y of the Euclidean building C ( X D , r , p ) is thic k . No w w e ma y apply Corollary 5.2 to the con tinuous map C ( f ). ✷ No w we pr o ve that a coarse equiv alence of Euclidean b uildings induces an isomorphism b et we en the spherical bu ildings at inﬁnit y . In order to sh o w th is, we ha ve to describ e the sph erical building at inﬁnity b y coarse data. 5.14 Lemma Let A 1 , . . . , A k b e aﬃne apartments in a Euclidean building X . The follo w in g are equiv alen t. (i) ∂ A 1 ∩ · · · ∩ ∂ A k 6 = ∅ . (ii) T here is an unb ounded set wh ic h is dominated by eac h of the aﬃn e apartments A 1 , . . . , A k . Pr o of. T o see that (i) implies (ii), let a ⊆ X b e a W eyl simplex represen ting an elemen t ∂ a ∈ ∂ A 1 ∩ · · · ∩ ∂ A k . F or eac h j there is a W eyl sim plex a j ⊆ A j represent ing ∂ a . Sin ce a and a j ha v e ﬁnite Hausdorﬀ distance, eac h A j dominates a . Before we p ro v e the con v erse implication, we note the follo wing. If an aﬃn e apartment A dom- inates a set Z ⊆ X , then π A Z h as ﬁnite Hausd orﬀ distance f rom Z . If Z is unb ounded, then ther e exists a W eyl simplex a ⊆ A th at con tains an un b ound ed subs et of π A Z . In particular, there exists then a W eyl simplex a ⊆ A that d ominates an u n b oun ded su bset of Z (or π A Z ). No w we assume that the un b oun ded s et Z ⊆ X is dominated b y the aﬃne apartments A 1 , . . . , A k . Consider the un b ounded set Y = π A 1 Z . T here exists a W eyl simplex a ⊆ A 1 of minimal dimension whic h dominates an unb ounded su bset Y 1 ⊆ Y . W e cla im that ∂ a ∈ ∂ A 1 ∩ · · · ∩ ∂ A k . Let j > 1. In A j w e ﬁnd a W eyl cham b er c j whic h dominates an un b ound ed subset Y j of π A j ( Y 1 ). Let A ′ j b e an aﬃne apartmen t cont aining representa tiv es a ′ and c ′ j of ∂ a and ∂ c j . Then Y ′ j = π A ′ j ( Y j ) has ﬁnite Hausd orﬀ d istance f rom Y j . Since b oth a ′ and c ′ j dominate the unb ounded s et Y ′ j , and since a ′ is also a W eyl simp lex of m inimal dimension wh ic h d ominates an unb ounded subset of Y , we ha v e ∂ a ⊆ ∂ c j ∈ ∂ A j . ✷ Supp ose that X and Y are E uclidean b uildings and that f : X × R m 1 ✲ Y × R m 2 is a coarse equiv alence. W e ma y view R m 1 as a E uclidean b uilding. Th en ev ery aﬃne apartment in X × R m 1 is of the form A × R m 1 , where A ⊆ X is an aﬃne apartmen t. By 5.9 there is a map f ∗ from the set of all aﬃn e apartment s of X to the set of all aﬃne apartmen ts of Y , su c h that f ( A × R m 1 ) has ﬁnite Hausdorﬀ distance from f ∗ A × R m 2 . If g is a coa rse inv erse of f , then g ∗ is an in v erse of f ∗ . 5.15 Lemma Sup p ose that X and Y are Euclidean buildings. Let f : X × R m ✲ Y × R m 38 b e a coarse equiv alence. Then f ∗ induces an isomorp h ism b et w een th e apartment complexes AC ( ∂ cpl X ) and AC ( ∂ cpl Y ) . Pr o of. Let A 1 , . . . , A k ⊆ X b e aﬃn e apartments with ∂ A 1 ∩ · · · ∩ ∂ A k 6 = ∅ . Let a b e a 1- dimensional W eyl simplex representing a vertex ∂ a ∈ ∂ A 1 ∩ · · · ∩ ∂ A k . Cho ose s > 0 such that a ⊆ B s ( A 1 ) ∩ · · · ∩ B s ( A k ). Let r f > 0 b e as in 5.9 and let Z = B r f + ρ ( s ) ( f ∗ A 1 ) ∩ · · · ∩ B r f + ρ ( s ) ( f ∗ A k ) ⊆ Y , where ρ is the control fun ction for f . Then f r estricts to a qu asi-isometric embedd ing of a × R m in to Z × R m . If Z is b ounded, w e get a qu asi-isometric embedd ing of a × R m ∼ = [0 , ∞ ) × R m in to R m , wh ic h is imp ossib le (b y top ological dimension in v ariance, applied to the asymptotic cones, see 1.17 and the pro of of 2.16). Hence Z is u n b oun ded. T herefore { f ∗ A 1 , . . . , f ∗ A k } is a simplex in AC ( ∂ cpl X 2 ) by 5.14. It follo ws that f ∗ is a simplicial map f ∗ : AC ( ∂ cpl X 1 ) ✲ AC ( ∂ cpl X 2 ). If g is a coarse in verse of f , then g ∗ is a simplicial inv erse of f ∗ . ✷ Com bining these results, we ha v e the follo wing ﬁr st main result about coarse equiv alences b etw een Euclidean buildings. T his is Theorem I I in the introdu ction. 5.16 The orem Let X 1 and X 2 b e Euclidean buildin gs wh ose spher ical buildings at inﬁnit y ∂ cpl X 1 and ∂ cpl X 2 are thic k. Let f : X 1 × R m 1 ✲ X 2 × R m 2 b e a coarse equiv alence. T h en m 1 = m 2 and the map f ∗ on the aﬃn e apartments extend s uniquely to a simplicial isomorphism f ∗ : ∂ cpl X 1 ✲ ∂ cpl X 2 . Pr o of. By 5.13 w e hav e m 1 = m 2 . By 5.15 the map f ∗ induces a simp licial isomorphism b et wee n AC ( ∂ cpl X 1 ) and AC ( ∂ cpl X 2 ). By 3.10 f ∗ induces a simplicial isomorph ism f ∗ : ∂ cpl X 1 ∼ = ✲ ∂ cpl X 2 . ✷ The thickness of the spherical bu ildings is essen tial f or th e argument. Ho wev er, th e Eu clidean factors whic h are allo we d in the theorem lead also to a result f or the case that the spherical bu ildings at inﬁnity are w eak build ings. 5.17 C orollary Let f : X 1 ✲ X 2 b e a coarse equ iv alence b etw een Eu clidean buildings. T hen the induced map on the aﬃne apartments induces an isomorphism b etw een the thic k b u ilding f actors in the reductions of ∂ cpl X 1 and ∂ cpl X 2 . Pr o of. Th is f ollo ws from 4.12, and 5.16 . ✷ Note that we do not claim that f ind uces directly a map ∂ f : ∂ X 1 ✲ ∂ X 2 b et w een th e Tits b ound aries in the sense of C A T(0) geometry . This w ill in general not b e the case; for example, f could b e a bi-Lipsc hitz homeomorphism of th e Euclidean cone EB( B ) o ve r a thic k sp herical bu ilding B . Suc h a self homeomorphism can b e rather wild at inﬁn it y . Our com b inatorial construction of f ∗ applies neve rtheless. 5.18 It remains to pr o ve Theorem I I I. W e ﬁx some notation. W e assume that X 1 and X 2 are metrically complete Euclidean build ings whose spherical buildings at inﬁnit y ∂ cpl X 1 and ∂ cpl X 2 are thic k . F urthermore, we assume that f : X 1 × R m 1 ✲ X 2 × R m 2 39 is a coa rse equiv alence. By 5.16, f in duces an isomorphism f ∗ : ∂ cpl X 1 ∼ = ✲ ∂ cpl X 2 whic h is c haracterized b y the fact that for ev er y aﬃn e apartmen t A ⊆ X 1 , the image f ( A × R m 1 ) has ﬁnite Hausdorﬀ distance from f ∗ A × R m 2 . W e p ut n = d im( X 1 ) = dim( X 2 ) . The Euclidean factors h a ve by 5.13 the same dimension, wh ic h w e denote by m = m 1 = m 2 . Finally , we pu t f ( x × v ) = f 1 ( x × v ) × f 2 ( x × v ) . Let a, b ∈ ∂ cpl X 1 b e opp osite panels. As in 4.24 , w e den ote b y X 1 ( a, b ) the union of all aﬃn e apartments in X 1 whose b ound ary con tains a and b . C onsider the closed con v ex su bsets Y 1 = X 1 ( a, b ) × R m ⊆ X 1 × R m and Y 2 = X 2 ( f ∗ a, f ∗ b ) × R m ⊆ X 2 × R m . The comp osite π Y 2 ◦ f : Y 1 ✲ Y 2 is a con tr olled map. I f A ⊆ X 1 ( a, b ) is an aﬃne apartmen t, then f ∗ A is an aﬃne apartment in X 2 ( f ∗ a, f ∗ b ); see 5.15. By 5.9, there is a uniform constan t r f > 0 su c h d ( f ( y ) , π Y 2 ( f ( y )) ≤ r f for all y ∈ Y 1 , so f | Y 1 and π Y 2 ◦ f | Y 1 ha v e ﬁ nite distance. If g is a coarse inv erse of f , then π Y 1 ◦ g | Y 1 is th erefore a coarse in v erse of π Y 2 ◦ f | Y 2 , and we obtain a coarse equiv alence π Y 2 ◦ f : Y 1 ✲ Y 2 . By 4.24 , Y i factors for i = 1 , 2 as a metric pr o duct T i × R n − 1+ m of a metrica lly complete leaﬂess tree T i , the w all tree, and a Eu clidean space. The ends of the wa ll tree T 1 corresp ond bijectiv ely to the c ham b ers of ∂ cpl X 1 con taining the panel a . If the t yp e of th e panel a is not isola ted in the Coxe ter diagram of ∂ cpl X 1 , then Π + ∂ cpl X 1 ( a ) acts 2-transitiv ely on the ends of T 1 . The b oun dary map f ∗ : ∂ cpl X 1 ✲ ∂ cpl X 2 is clearly equiv arian t with r esp ect to the isomo rph ism Π + ∂ cpl X 1 ( a ) ✲ Π + ∂ cpl X 2 ( f ∗ a ). I n this situatio n we ma y apply 2.18 and conclude that f ∗ extends to an isometry from T 1 to T 2 , p ossibly after rescaling. 5.19 Lemma Assume that f : X 1 × R m ✲ X 2 × R m is as in 5.18. If ∂ cpl X 1 is irredu cible and dim( X 1 ) = n ≥ 2 and if some w all tree of X 1 is of t yp e (I), then there is an isometry ¯ f : X 1 ✲ X 2 with ( ¯ f × id R m ) ∗ = f ∗ . Pr o of. If the wall tr ee T 1 is of t yp e (I), then X 1 = EB( ∂ cpl X 1 ) by 4.26. S ince ∂ cpl X 1 is th ic k and irreducible, Π + ∂ cpl X 1 ( a ) acts 2-transitiv ely on the ends of T 1 b y 3.14. By 2.18, the wall trees of X 2 are also of t yp e (I). Therefore X 2 = EB( ∂ cpl X 2 ). The isomorphism f ∗ : ∂ cpl X 1 ✲ ∂ cpl X 1 extends to an isometry ¯ f : X 1 ✲ X 2 of the resp ectiv e Euclidean cones. ✷ 5.20 Lemma Assume that f : X 1 × R m ✲ X 2 × R m 40 is as in 5.18. If ∂ cpl X 1 is irr educible and dim( X 1 ) = n ≥ 2 and if no wall tree o f X 1 is of t yp e (I), then the metric on X 2 can b e rescaled so that there is an isometry ¯ f : X 1 ✲ X 2 with ( ¯ f × id R m ) ∗ = f ∗ . Th is isometry ¯ f is uniqu e and there is a constan t r > 0 such that d ( f 1 ( x × v ) , ¯ f ( x )) ≤ r for all x × v ∈ X 1 × R m . Pr o of. Let ( a, b ) b e a pair of opp osite panels in ∂ cpl X 1 . Then ( a, b ) determines a wal l (a sphere of d imension n − 2) in the spherical b uilding ∂ cpl X 1 . ∂ cpl X 1 is ir reducible, there are at most 2 typ es of wa lls in ∂ cpl X 1 . The group Π + ∂ cpl X 1 ( a ) acts 2-transitiv ely on the ends of the w all tree T 1 b y 3.14. W e ma y apply 2.18 to the coarse equiv alence T 1 × R n − 1+ m ✲ T 2 × R n − 1+ m , since the equiv ariance co ndition is satisﬁed by our pr evious discussion. O nce and for all, w e r escale the metric on X 2 in such a wa y that f ∗ extends to an equiv ariant isometry τ : T 1 ✲ T 2 . If ( a ′ , b ′ ) is an y other pair of panels in ∂ cpl X 1 of the same type a s ( a, b ), with X 1 ( a ′ , b ′ ) = T ′ 1 × R n − 1 , then there is some pro j ectivit y wh ic h induces an isometry ϕ 1 : T 1 ✲ T ′ 1 . P ushing this pro jectivit y forw ard via f ∗ , w e obtain an isometry ϕ 2 : T 2 ✲ T ′ 2 , where X 2 ( f ∗ a ′ , f ∗ b ′ ) = T ′ 2 × R n − 1 . By construction, the maps ϕ 2 ◦ τ ◦ ϕ − 1 1 and f ∗ induce the same map ∂ T ′ 1 ✲ ∂ T ′ 2 . By 2.18, the map ϕ 2 ◦ τ ◦ ϕ − 1 1 is the unique equiv arian t tree isometry which accompanies the coarse equiv alence X 1 ( a ′ , b ′ ) ✲ X 2 ( f ∗ a ′ , f ∗ b ′ ). T his shows that with resp ect to our m etrics on X 1 and X 2 , the map f ∗ : ∂ cpl X 1 ✲ ∂ cpl X 2 is ecologica l for all wall trees whose wa ll in ∂ cpl X 1 is of the same t yp e as the w all determined by ( a, b ). Supp ose that z ∈ X 1 is a thic k p oint in an aﬃne apartment A ⊆ X 1 . Sin ce X 1 is irreducible, z is the intersectio n of n wa lls M 1 , . . . , M n ⊆ A whic h are of the same t yp e as the wa ll determined b y ( a, b ). T o see this, w e note that the W -orbit of an y nonzero v ector spans the am bien t Euclidean space. Eac h of these wal ls determines a b ranc h p oin t in a panel tree. Let M 1 , ∗ , . . . , M n, ∗ b e the corresp onding wa lls in f ∗ A , deﬁned by the isometries b etw een the corresp onding wall trees. The in tersection M 1 , ∗ ∩ · · · ∩ M n, ∗ is a p oin t z ∗ ∈ f ∗ A . By 2.18 there is a uniform constan t s > 0 suc h that d ( f 1 ( z × v ) , z ∗ ) ≤ s . If z ′ ∈ A is another thic k p oint and if z ′ ∗ is constructed in the s ame wa y , then the n tree isometries y ield d ( z , z ′ ) = d ( z ∗ , z ′ ∗ ), whence d ( f 1 ( z × v ) , f 1 ( z ′ × v ) ≤ d ( z , z ′ ) + 2 s . The thic k p oin ts are cob ound ed in X 1 , so the map x 7− → f 1 ( x × v ) is a rough isometry X 1 ✲ X 2 . This implies by 2.18 that for n o wal l tree of X 1 , the accompan ying isometry requires any rescaling of X 2 . The acco mpanying tree isometries therefore ﬁt to gether with f ∗ to an eco logical bu ilding isomorphism. By Tits’ result 4.28, f is acco mpanied b y an isometry ¯ f : X 1 ✲ X 2 . Th is isometry maps the thic k p oint z p recisely to the p oin t z ∗ describ ed ab o ve , z ∗ = ¯ f ( z ). It follo ws that there is a constan t r > 0 suc h that d ( f 1 ( x × v ) , ¯ f ( x )) ≤ r holds for all x × v ∈ X 1 × R m . ✷ W e n o w decomp ose the E uclidean b uilding X 1 in to a pro du ct X I 1 × X I I 1 × X I I I 1 of Eu clidean bu ildings of t yp es (I), (I I) and (I I I), resp ectiv ely . Sim ilarly , we deco mp ose X 2 . Th e next result implies Theorem I I I. 5.21 The orem Let f : X 1 × R m 1 ✲ X 2 × R m 2 b e a coarse equiv alence of Eu clidean buildin gs. Assum e that ∂ cpl X 1 and ∂ cpl X 2 are thick and that X 1 splits oﬀ no tree factors. Then th e follo win g hold. 41 (o) T here are n um b ers m, n with m 1 = m 2 = m and dim( X 1 ) = dim( X 2 ) = n . (i) The irreducible factors of X 2 can b e rescaled in su c h a w a y that there is an isometry ¯ f : X 1 ✲ X 2 with f ∗ = ( ¯ f × id R m ) ∗ . (ii) I f X 1 = X I 1 × X I I 1 × X I I I 1 and X 2 = X I 2 × X I I 2 × X I I I 2 are decomp osed as ab o v e, then ¯ f factors as a pro duct, ¯ f ( x I × x I I × x I I I ) = ¯ f I ( x I ) × ¯ f I I ( x I I ) × ¯ f I I I ( x I I I ) . There is a constan t r > 0 such that d ( ¯ f N ( x N ) , π X N f ( x I × x I I × x I I I × p )) ≤ r , for N ∈ { I I , I I I } and f or all x I × x I I × x I I I × v ∈ X I 1 × X I I 1 × X I I I 1 × R m . (iii) The maps f I I and f I I I are unique. Pr o of. W e pro ceed by induction on the num b er irreducible factors of X 1 . The case of one irreducible factor is co vered by 5.19 and 5.20. In general, we d ecomp ose X 1 = Y 1 × Z 1 , with Z 1 irreducible. As f ∗ is an isomorp hism, we ha v e a corresp ond ing decomp osition X 2 = Y 2 × Z 2 . Fix opp osite c hambers a, b in ∂ cpl Y 1 , and let A ⊆ Y 1 b e the corresp onding aﬃn e apartmen t. T hen A × Z 1 is th e un ion of all aﬃne apartments in Y 1 × Z 1 whic h conta in a, b at inﬁ nit y . T his relation is pr eserv ed b y f ∗ . So if y × z × v ∈ A × Z 1 × R m , then f ( y × z × v ) has un iform distance from f ∗ A × Z 1 × R m . It f ollo ws that π f ∗ A × Z 2 × R m ◦ f is a coarse equiv alence b et ween A × Z 1 × R m and f ∗ A × Z 2 × R m . By the induction hyp othesis we can rescale the irred ucible factors of Y 2 in suc h a w a y that there is an isometry b et w een Y 1 and Y 2 , and 5.19 and 5.20 giv e us isometries b et w een Z 1 and Z 2 , p ossibly after rescaling Z 2 . These isometries ﬁt toge ther to an isometry ¯ f : X 1 ✲ X 2 , with f ∗ = ¯ f ∗ . This giv es (i). If Z 1 is of type (I I) or (I I I ), then the claims (ii) and (iii) follo w, by applying 5.20 to A × Z 1 × R m and f ∗ A × Z 2 × R m . ✷ 42 App endix to: ‘Coarse equiv alence s of Euc lidean building s’ b y Jero en S c hillew aert ∗ and Ko en Stru yv e † The pu rp ose of this app endix is to pr o vid e a generaliza tion of the m ain r esults of the p receding pap er ‘Coarse equiv alences of Euclidean buildings’ by Linus Kramer and Ric hard W eiss. W e will refer to this pap er as [KW]. Th e pap er in question pr o ves rigidit y results for coarse equiv alences of metrically complete Euclidean buildings. W e will show that the hypothesis of completeness can b e omitted. The pr o of for this generalizati on is ce rtainly not indep enden t of the original p ro of. In fact, we only discuss wh ere a nd ho w the pr o of of Kramer a nd W eiss should b e a ltered to a llo w for non-complete Euclidean build ings. As our p ro of is an extension of the Kramer and W ei ss argumen t, w e r efer and rely on it for a d etaile d in tro d uction, details and deﬁn itions. Here are the theorems that we obtain. Theorem A.I L et G b e a gr oup acting isometric al ly on two le aﬂess tr e es T 1 , T 2 . Assume that ther e is a c o arse e quivalenc e f : T 1 ✲ T 2 , that T 1 has at le ast 3 ends and that the induc e d map ∂ f : ∂ T 1 ✲ ∂ T 2 b etwe en the ends of the tr e es is G -e q u ivariant. If the G -action on ∂ T 1 is 2 -tr ansitive, then (after r esc aling the metric on T 2 ) ther e is a G -e quiv ariant isometry ¯ f : T 1 ✲ T 2 with ∂ f = ∂ ¯ f . If T 1 has at le ast two br anch p oints, then ¯ f is unique and has ﬁnite distanc e fr om f . Theorem A.I I I L et f : X 1 × R m ✲ X 2 × R m b e as in The or em II in [KW] and assume in addition that X 1 has no tr e e factors. Then ther e i s (p ossibly after r esc aling the metrics on the de Rh am factors of X 2 ) an isometry ¯ f : X 1 ✲ X 2 with ( ¯ f × id R m ) ∗ = f ∗ . Pu t f ( x × v ) = f 1 ( x × v ) × f 2 ( x × v ) . If none of the de Rham factors of X 1 is a Euclide an c one over its b oundary, then ¯ f is uniqu e and d ( f 1 ( x × y ) , ¯ f ( x )) is b ounde d as a function of x ∈ X 1 . W e no w discuss the mo diﬁcations one has to make in order to a v oid the assumption of metrical com- pleteness in the results on coarse rigidit y of Lin us Kramer and Ric hard W eiss in [KW]. C ompleteness is u sed at tw o places in their p ro of, wh ic h we will d iscuss separately . All references are to [KW] unless men tioned otherwise. Reco v ering the tree from the G -action The ﬁr st p roblem that o ccur s is that the Bruh at-Tits Fixed Poin t Theorem r equires completeness, so a b ounded isometry group act ing on a metrically non-complete R -tree T d o es n ot n ecessarily h a ve a ﬁxed p oint (although it do es in th e metric completion T of T ). Consequ en tly also Prop osition 2.8 no longer holds. The next lemma and prop ositions oﬀer a w ay to deal with this observ ation. F or a geod esic segmen t [ x, y ] in a tree we p ut ( x, y ) = [ x, y ] − { x, y } . A.1 Lemma Let T b e an R -tree and x , y ∈ T . Then the op en segment ( x, y ) lies in T . ∗ The ﬁ rst author is supp orted by Marie Curie IEF Grant GELA TI (EC grant nr 328178) † The second author is su pp orted by th e F und for S cien tiﬁc Researc h — Flanders (FWO - Vlaanderen) 43 Pr o of. First of all note that th e metric completio n of an R -tree is again an R -tree by [23] (see also [33, Cor. I I.1.10]), hence it mak es sense to sp eak about the op en segmen t ( x, y ). (Note how ev er that the metric completion of a leaﬂess metrically non-complete R -tree is nev er leaﬂess.) Giv en ε > 0, w e c ho ose p oints x ′ , y ′ ∈ T with d ( x, x ′ ) , d ( y , y ′ ) < ε . The geodesic segmen t [ x ′ , y ′ ] is conta ined in T . W e put x ′′ = π [ x ′ ,y ′ ] ( x ) and y ′′ = π [ x ′ ,y ′ ] ( y ). Th e unique geod esic from x to x ′ then passes thr ough x ′′ . In particular, d ( x, x ′′ ) < ε , and similarly d ( y , y ′′ ) < ε . By (T2), we ha ve that [ x, y ] = [ x, x ′′ ] ∪ [ x ′′ , y ′′ ] ∪ [ y ′′ , y ]. The claim follo ws no w , b ecause [ x ′′ , y ′′ ] ⊆ T and b ecause ε w as arbitrarily small. ✷ F or the p urp ose of th is app endix, we restate cond ition (2- ∂ ) without the completeness assump tion: (2- ∂ ) The group G acts isometric ally on th e leaﬂess tree T , and this action is 2-transitiv e on ∂ T . The next t w o p rop ositions serv e as replacement s of Prop ositions 2.7 and 2.8 of [KW]. A.2 Prop osition Assu m e that the tree T s atisﬁes (2- ∂ ) and that the s et of branch p oints is dense. Then every p oint x ∈ T is G -isolated in T . Pr o of. L et x ∈ T . Su pp ose that G x ﬁxes another p oin t y 6 = x in T . Let y ′ ∈ ( x, y ). Then y ′ lies in T by Lemm a A.1 and G x ﬁxes the geo desic seg ment [ x, y ′ ]. Th ere is a br anc h p oin t z b etw een x and y ′ , so Prop osition 2.3 implies th at y ′ is n ot a ﬁxed p oint of G x,z . ✷ This mo diﬁcation together with the Bruh at-Tits Fixed P oint Theorem allo ws us, as in [KW], to conclude that the stabilizers of G -isolated p oin ts in T are maximal b ounded su bgroups. A.3 Prop osition Assu m e (2 - ∂ ) and that P ⊆ G is a m aximal b ound ed subgroup. Then P is the stabilizer of a G -isolated p oin t, or it only ﬁxes exactly one p oin t in th e completion T and none in T . Pr o of. Note that w e ma y assume that the tree is metrically non-complete, and h ence that the tree T is of t yp e (I I I) with a d ense s et of b ranc h p oints (see Corollary 2.6). Assume that what we wan t to prov e is false. As in Prop osition 2.8, this imp lies that T P con tains a ge o desic segment [ x, y ] ⊆ T with x 6 = y . Applyin g Lemm a A.1 one has that ( x, y ) ⊂ T , so there exists a sub segmen t [ x ′ , y ′ ] ⊂ ( x, y ) ⊂ T P with x ′ 6 = y ′ . Th is situation w as pro ved to b e imp ossible in Prop osition 2.8. ✷ W e are still left with the p ossib ilit y that ther e are more maximal b ounded subgroup s th an only those corresp onding to G -isolated p oints of T . Ho wev er those corresp onding to G -isolate d p oin ts can b e recognized as follo ws : A.4 Lemma A maximal b ound ed subgroup H is the stabilizer of a G -isolated p oin t of T if and only if either T is n ot of t yp e (I I I), or if T is of t yp e (I I I) and for eac h tw o ends u, v ∈ ∂ T , H is the only maximal b ound ed subgroup of G con taining h ( G u ∪ G v ) ∩ H i . Pr o of. Let H b e a maximal b oun ded subgroup and let x b e the p oin t stabilized by H . If the tree is not of typ e (I I I), then th e tree is metrically complete and no extra maximal b ounded sub groups app ear. If the tree is of t yp e (I I I) then Lemma 2.11 states th at for eac h tw o ends u, v ∈ ∂ T , the group H is th e only m aximal b ounded subgroup of G con taining h ( G u ∪ G v ) ∩ H i if and on ly if x lies in some apartmen t. This h olds if and only if x is G -isola ted. ✷ 44 Note that one can still distinguish b et ween typ es (0)-(I I I) using Lemm a A.1 and 2.12. No w we can mo d ify th e last p art of Section 2 (starting directly after Pr op osition 2.8), and d eﬁne i G ( T ) to corresp ond with th e set of maximal b ounded su bgroups of G satisfying th e conditions of the ab o v e lemma. W e conclude that P rop osition 2.13 still holds in th e metrically n on-complete case. Similarly one h as to alter Pr op osition 2.17 and Th eorem 2.18 restricting to the maximal b ound ed subgroups with the ab ov e p rop erty . Ob serv e that this pr op ert y is preserv ed by the G -e quiv ariance. The use of retraction maps π In 1.1 one r efers to [7, I I.2.4] for a 1-Lipsc hitz retraction map π K : X ✲ K when K is a metrically complete conv ex subset in a CA T(0)-space X . This kind of retraction maps is us ed in v arious places of [KW]. In most of these places, K is a closed conv ex su bset of an apartmen t and there is n o problem. The only place wh ere this is not th e case is at the b eginning of 5.18. 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Lin us Kramer Mathematisc hes Institut, Unive rsit¨ at M ¨ u nster, Einsteinstr. 62, 48149 M¨ un ster, Germany e-mail: linu s.kramer @uni-muen ster.de Jero en Sc hillew aert Departmen t of Mathematics, Imp erial College Lon don, South K ensington Campus , London SW7 2AZ, England e-mail: jschillew aert@gmail.com Ko en Struyve Departmen t Mathematics, Ghent Universit y , Krijgslaan 281, S22, B-9000 Ghent, Belgium e-mail: kstr uyve@cag e.ugent.b e Ric h ard M. W eiss Dept. of Mathematics, T u fts Un iv ersit y , 503 Boston Av e., Medford, MA 02155, USA e-mail: rich ard.weis s@tufts.e du 48

Coarse Equivalences of Euclidean Buildings

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