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RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS by BRuc~, KLEINER* and BERNHARD LEEB** 1. INTRODUCTION 1.1. Background and statement of results An (L, C) quasi-isometry is a map (I) : X --~ X' between metric spaces such that for all xx, x~ e X we have (I) L -1 d(x:, xz) -- C .< d((I)(x:), (I)(xz)) .< L d(x:, x2) + C and d(x', Im((I))) < C (2) for all x' e X'. Quasi-isometrics occur naturally in the study of the geometry of discrete groups since the length spaces on which a given finitely generated group acts cocompactly and properly discontinuously by isometrics are quasi-isometric to one another [Gro]. Quasi-isometrics also play a crucial role in Mostow's proof of his rigidity theorem: the theorem is proved by showing that equivariant quasi-isometrics are within bounded distance of isometrics. This paper is concerned with the structure of quasi-isometries between products of symmetric spaces and Euclidean buildings. We recall that Euclidean spaces, hyperbolic spaces, and complex hyperbolic spaces each admit an abundance of self-quasi-isometries [Pan]. For example we get quasi-isometries EZ--~ E 2 by taking shears in rectangular (Xl, X2) I--r (Xl, X 2 +f(xl) ) or polar (r, O) ~-, (r, 0 +f(r)/r) coordinates, where f: R ~ R and g : [0, oo) --> R are Lipschitz. Any diffeomorphism (1) (I) : 0H" -+ 0H" of the ideal boundary can be extended continuously to a quasi-isometry (I):H"-+ H ". Likewise any contact diffeomorphism (3) 0(I) : 0CH" -+ 0CH" can be extended continuously to * The first author was supported by NSF and MSRI Postdoctoral Fellowships and the Sonderforschungs- bereich SFB 256 at Bonn. ** The second author was supported by an MSRI Postdoctoral Fellowship, the SFB 256 and IHES. (I) Any quasi-conformal homeomorphism arises as the boundary homeomorphism ofa quasi-isometry by [Tuk]. (2) The boundary of CH n can be endowed with an Isom(CH n) invariant contact structure by projecting the contact structure from a unit tangent sphere ~1~2n--1 CH n to ~CH n using the exponential map. 116 BRUCE KLEINER AND BERNHARD LEEB a quasi-isometry O:CH". CH ~ [Pan]. Quasi-isometrics of the remaining rank 1 symmetric spaces of noncompaet type, on the other hand, are very special. They are essentially isometries: Theorem 1.1.1 ([Pan]). -- Let X be either a quaternionic hyperbolic space Ill-I", n > 1, or the Cayley hyperbolic plane Call 2. Then any quasi-isometry Of X lies within bounded distance of an isometry. Note that Pansu's theorem is a strengthening of Mostow's rigidity theorem for these rank one symmetric spaces X, as it applies to all quasi-isometrics of X, whereas Mostow's argument only treats those quasi-isometries which are equivariant with respect to lattice actions. The main results of this paper are the following higher rank analogs of Pansu's theorem. Theorem 1.1.2 (Splitting). -- For 1 <~ i <~ k, 1 <~ j ~ k' let each X~, X; be either a nonflat irreducible symmetric space of noncompact type or an irreducible thick Euclidean Tits building with cocompact affine Weft group (see section 4.1 for the precise definition). Let k ~ t X = 1r � II~=i X~, X' = E"' � 1-I~=1 Xj be metric products (x). Then for every L, C there are constants L, C and f) such that the following holds. If ~ : X ~ X' is an (L, C) quasi- isometry, then n = n', k = k', and after reindexing the factors of X' there are (L, C) quasi- isometries ~i:X~-+X" so that d(p'oCb, IIrb~op)<D, where p:X~11 ~=lx~ and p' : X' -+IP ~= ~ X~ are the projections. A more general theorem about quasi-isometries of products is proved in [KKL]. Theorem 1.1.3 (Rigidity). -- Let X and X' be as in Theorem 1.1.2, but assume in addition that X is either a nonflat irreducible symmetric space of noncompact type of rank at least 2, or a thick irreducible Euclidean building of rank at least 2 with cocompact affine Weyl group and Moufang Tits boundary. Then any (L, C) quasi-isometry rb : X -~ X' lies at distance < D from a homothety cb o : X ~ X', where D depends only on (L, C). Theorem 1.1.3 settles a conjecture made by Margulis in the late 1970's, see [Gro, p. 179] and [GrPa, p. 73]. It is shown in [Le] that the Moufang condition on the Tits boundary of X can be dropped. As an immediate consequence of Theorems 1.1.2, 1.1.3, and [Mos] we have: Corollary 1.1.4 (Q2~asi-isometric classification of symmetric spaces). -- Let X, X' be symmetric spaces of noncompact type. If X and X' are quasi-isometric, then they become isometric after the metrics on their de Rham factors are suitably renormalized. Mostow's work [Mos] implies that two quasi-isometric rank 1 symmetric spaces of noncompact type are actually isometric (up to a scale factor) ; and it was known by [AS] that two quasi-isometric symmetric spaces of noncompact type have the same rank. We will discuss other applications of Theorems 1 1.2 and 1.1.3 elsewhere, see [K1Le2] and [KILe3]. (x) The distance function on the product space is given by the Pythagorean formula. RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 117 1.2. Commentary on the proof Our approach to Theorems 1.1.2 and 1.1.3 is based on the fact that if one scales the metrics on X and X' by a factor ~, then (L, C) quasi-isometries become (L, XC) quasi-isometries. Starting with a sequence X~ --~ 0 we apply the ultralimit construction of [DW, Gro] to take a limit of the sequence 9 : ~ X --~ X~ X', getting an (L, 0) quasi- isometry (i.e. a biLipschitz homeomorphism) ~,0 : X~ ~ X~ between the limit spaces. The first step is to determine the geometric structure of these limit spaces: Theorem 1.2.1. -- The spaces X,~ and X~ are thick (generalized) Euclidean Tits buildings (cf. section 4.1). The second step is to study the topology of the Euclidean buildings X~, X~,. We establish rigidity results for homeomorphisms of Euclidean buildings which are topological analogs of Theorems 1.1.2 and 1.1.3: Theorem 1.2 2. -- Let Y~, Y~ be thick irreducible Euclidean buildings with topo- logically transitive affine Weyl group (cf. section 4.1.1), and let Y ---- E"� I] k~=l Y~, t t ! Y' -~ E"' � H~'= 1 Yj. If ~ : Y -+ Y' is a homeomorphism, then n = n, k = k, and after reindexing factors there are homeomorphisms tF, : Yi ~ Y~ so that p' o 9 -~ l-I~i o p where p : y -+ I-[ k p' y' -+ IIk , = 1 Y~ and : ~ = 1 Y~ are the projecticns. Theorem 1.2 3. -- Let Y be an irreducible thick Euclidean building with topologically transitive affine Weyl group and rank >t 2. Then any homeomorphism from Y to a Euclidean building is a homothety. For comparison we remark that if Y and Y' are thick irreducible Euclidean buildings with crystallographic (i.e. discrete cocompact) affine Weyl group, then one can use local homology groups to see that any homeomorphism carries simplices to simplices. In particular, the homeomorphism induces an incidence preserving bijection of the simplices of Y with the simplices of Y', which easily implies that the homeo- morphism coincides with a homothety on the 0-skeleton. In contrast to this, homeo- morphisms of rank 1 Euclidean buildings with nondiscrete affine Weyl group (i.e. R-trees) can be quite arbitrary: there are examples of R-trees T for which every homeomorphism A ~ A of an apartment A C T can be extended to a homeomorphism of T. However, we always have: Proposition 1.2.4. -- If X, X' are Euclidean buildings, then any homeomorphism ~:X ~ X' carries apartments to apartments. In the third step, we deduce Theorems 1 1 2 and 1. I. 3 from their topological analogs. By using a scaling argument and Proposition 1 2 4 we show that if X and X' are as in Theorem 1 1 2, and (I) : X -+ X' is an (L, C) quasi-isometry, then the image of a maximal fiat in X under (I) lies within uniform Hausdorff distance of a maximal 118 BRUCE KLEINER AND BERNHARD LEEB fiat in X'; the Hausdorff distance can be bounded uniformly by (L, C). In the case of Theorem 1.1.2 we use this to deduce that the quasi-isometry respects the product structure, and in the case of Theorem 1.1.3 we use it to show that 9 induces a well- defined homeomorphism O~:0X--> 0X' of the geometric boundaries which is an isometry of Tits metrics. We conclude using Tits' work [Til] (as in [Mos]) that 0~ is also induced by an isometry ~0 : X ---> X', and d(~, ~0) is bounded uniformly by (L, C). The reader may wonder about the relation between Theorems 1.1.2 and 1.1.3 and Mostow's argument in the higher rank case. An important step in Mostow's proof shows that if F acts discretely and cocompactly on symmetric spaces X and X', then any F-equivariant quasi-isometry ~:X ~ X' carries maximal flats in X to within uniform distance of maximal fiats in X'. The proof in [Mos] exploits the dense collection of maximal flats with cocompact F-stabilizer (1). One can then ask if there is a " direct" argument showing that maximal flats in X are carried to within uniform distance of maximal fiats in X' by any quasi-isometry (2); for instance, by analogy with the rank 1 case one may ask whether any r-quasi-flat (3) in a symmetric space of rank r must lie within bounded distance of a maximal flat. The answer is no. If X is a rank 2 symmetric space, then the geodesic cone [J8 E s ~ over any embedded circle S in the Tits boundary 0~:lt . X is a 2-quasi-flat. Similar constructions produce nontrivial r-quasi-flats in sym- metric spaces of rank >/ 2. But in fact this is the only way to produce quasiflats: Theorem 1.2.5 (Structure of quasi-flats). -- Let X be as in Theorem 1.1.2, and let r = rank(X). Given L, C there are D, D' e N such that every (L, C) r-quasi-flat Qc x lies within the D-tubular neighborhood ND([.J~.e~- F) of a union of at most D maximal flats. Moreover, the limit set of Q is the union of at most D' closed Weyl chambers in the Tits boundary OTl ~ X. It follows easily that if L is sufficiently close to 1 (in terms of the geometry of the spherical Coxeter complex (S, W) associated to X) then any (L, C) r-quasi-fiat in X is uniformly close to a maximal flat. In the special case that X is a symmetric space, Theorem 1.2.5 was proved independently by Eskin and Farb, approximately one year after we had obtained the main results of this paper for symmetric spaces. We would like to mention that related rigidity results for quasi-isometries have been proved in [Sch]. 1.3. Org~nlzation of the paper Section 2 contains background material which will be familiar to many readers; we recommend starting with section 3, and using section 2 as a reference when needed. We provide the straight-forward generalization of some well-known facts about Hada- (1) If Zr C F acts cocompactly on a maximal flat F C X, then Zr will stabilize ~(F) and a flat F' in X'. One can then get a uniform estimate on the Hausdorff distance between ~(F) and F'. (2) Obviously this statement is true by Theorems I. 1.2 and 1.1.3. (s) An r-quasi-flat is a quasi-isometric embedding q~:Er----> X; a quasi-isometric embedding is a map satisfying condition (1), but not necessarily (2). RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS I I9 mard spaces to the non locally compact case. This is needed when we study the limit spaces X,~ which are non locally compact Hadamard spaces. Sections 3 and 4 give a self-contained exposition of the building theory used elsewhere in the paper. This exposition has several aims. First, we hope that it will make building theory more accessible to geometers since it is presented using the language of metric geometry, and we do not require any knowledge of algebraic groups. Second, it introduces a new definition of buildings (spherical and Euclidean) which is based on metric geometry rather than a combinatorial structure such as a polysimplicial complex. Tits' original definition of a building was motivated by applications to algebraic groups, whereas the objectives of this paper are primarily geometric. Here buildings (spherical and Euclidean) arise as geometric limits of symmetric spaces, and we found that the geometric definition in sections 3 and 4 could be verified more directly than the stan- dard one; moreover, the Euclidean buildings that arise as limits are " nondiscrete ", and do not admit a natural polysimplicial structure. Finally, sections 3 and 4 con- tain a number of new results, and reformulations of standard results tailored to our needs. Section 5 shows that the asymptotic cone of a symmetric space or Euclidean building is a Euclidean building. Section 6 discusses the topology of Euclidean buildings, proving Theorems t. 2.2, 1.2.3, 1.2.4. Section 7 proves that if X, X' and q) are as in Theorem 1.1.2, then the image of a maximal flat under q) is uniformly Hansdorff close to a flat (actually the hypotheses on X and X' can be weakened somewhat, see Corollary 7.1.5). General quasiflats are also studied in section 7; we prove there Theorem 1.2.5. Section 8 contains the proofs of Theorems 1.1.2 and 1.1.3, building on section 7. There is considerable overlap in the final step of the argument with [Mos] in the symmetric space case. 1.4. Suggestions to the reader Readers who are already familiar with building theory will probably find it useful to read sections 3.1, 3.2 and 4.1, to normalize definitions and terminology. The special case of Theorem 1.1.2 when X = X' = H 2 � H z already contains most of the conceptual difficulties of the general case, but one can understand the argument in this case with a minimum of background. To readers who are unfamiliar with asymptotic cones, and readers who would like to quickly understand the proof in a special case, we recommend an abbreviated itinerary, see appendix 9. In general, when the burden of axioms and geometric minutae seems overwhelming, the reader may read with the Rank 1 x Rank 1 case in mind without losing much of the mathe- matical content. 120 BRUCE KLEINER AND BERNHARD LEEB CONTENTS 1, Introduction ............................................................................. 1.1. Background and statement of results .................................................. 1.2, Commentary on the proof ........................................................... 1,3. Organization of the paper ........................................................... 1.4. Suggestions to the reader ............................................................ 2. Preliminaries ............................................................................. 2. I. Spaces with curvature bounded above ................................................ 2. I, I. Definition ................................................................... 2. 1.2. Coning ...................................................................... 2. I. 3. Angles and the space of directions of a CAT(x)-space ............................ 123 2.2. CAT(1)-spaces ...................................................................... 2.2. I. Spherical join ................................................................ 2.2.2. Convex subsets and their poles ................................................. 125 2,3. Hadamard spaces ................................................................... 2.3, I, The geometric boundary ...................................................... 2.3,2. The Tits metric .............................................................. 2.3.3. Convex subsets and parallel sets ............................................... 128 2.3.4. Products .................................................................... 2.3.5. Induced isomorphisms of Tits boundaries ....................................... 130 2.4. U1tralimits and asymptotic cones ..................................................... 131 2.4. I. Ultrafilters and ultralimits .................................................... 2.4.2. Ultralimits of sequences of pointed metric spaces ................................ 2.4.3. Asymptotic cones ............................................................. 133 3. Spherical buildings ........................................................................ 134 3. l. Spherical Coxeter complexes ......................................................... 134 3.2. Definition of spherical buildings ...................................................... 135 3.3. Join products and decompositions .................................................... 136 3.4. Polyhedral structure ................................................................. 138 3.5. Recognizing spherical buildings ...................................................... 3.6. Local conical@, projectivity classes and spherical building structure on the spaces of directions 140 3.7. Reducing to a thick building structure ................................................ 141 3.8. Combinatorial and geometric equivalences ............................................. 143 3.9. Geodesics, spheres, convex spherical subsets ............................................ 144 3.10. Convex sets and subbuildings ......................................................... 144 3.11. Building morphisms ................................................................. 3.12. Root groups and Moufang spherical buildings .......................................... 4. Euclidean buildings ....................................................................... 149 4, 1. Definition of Euclidean buildings ..................................................... 149 4.1. I. Euclidean Coxeter complexes .................................................. 149 4.1.2. The Euclidean building axioms ................................................ 150 4. 1.3. Some immediate consequences of the axioms .................................... 4.2. Associated spherical building structures ................................................ 151 4.2.1. The Tits boundary ........................................................... 151 4.2.2. The space of directions ....................................................... 152 4.3. Product (-decomposition) s ............................................................ 153 4.4. The local behavior of Weyl-cones .................................................... 154 4.4.1. Another building structure on ~]~ X, and the local behavior of Weyl sectors ........ 155 4.5. Discrete Euclidean buildings .......................................................... 156 RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 121 4.6. Flats and apartments ............................................................... 157 4.7. Subbuildings ....................................................................... 159 4.8. Families of parallel flats ............................................................. 160 4.9. Reducing to a thick Euclidean building structure ...................................... 161 4.10. Euclidean buildings with Moufang boundary .......................................... 164 5. Asymptotic cones of symmetric spaces and Euclidean buildings ................................ 167 5.1. Ultralimits of Euclidean buildings are Euclidean buildings ............................... 167 5.2. Asymptotic cones of symmetric spaces are Euclidean buildings ........................... 169 6. The topology of Euclidean buildings ....................................................... 171 6.1. Straightening simplices .............................................................. 172 6.2. The local structure of support sets .................................................... 173 6.3. The topological characterization of the link ........................................... 174 6.4. Rigidity of homeomorphisms ......................................................... 174 6.4.1. The induced action on links .................................................. 175 6.4.2. Preservation of fiats .......................................................... 175 6.4.3. Homeomorphisms preserve the product structure ................................ 175 6.4.4. Homeomorphisms are homotheties in the irreducible higher rank case ................ 176 6.4.5. The case of Euclidean de R_ham factors ........................................ 177 7. Q uasiflats in symmetric spaces and Euclidean buildings ...................................... 177 7.1. Asymptotic apartments are close to apartments ........................................ 177 7.2. The structure of quasi-flats .......................................................... 180 8. Q uasi-isometries of symmetric spaces and Euclidean buildings ................................. 185 8.1. Singular fiats go close to singular fiats ................................................ 185 8.2. Rigidity of product decomposition and Euclidean de Rham factors ...................... 187 8.3. The irreducible case ................................................................ 187 8.3.1. Quasi-isometries are approximate homotheties .................................... 188 8.3 2. Inducing isometries of ideal boundaries of symmetric spaces ....................... 190 8.3.3. (1, A)-quasi-isometries between Euclidean buildings .............................. 191 9. An abridged version of the argument ...................................................... 194 ................................................................................ 196 2. PRELIMINARIES Z. 1. Spaces with curvature bounded above General references for this section are [ABN, Ba, BGS]. 2.1.1. Definition If *: e R, let M~ be the two-dimensional model space with constant curvature ; let D(~r = Diam(M~). A complete metric space (X, [ . [) is a CAT(,:)-space if 1. Every pair xt, x~ E X with [ x I x 2 [ < D(K) is joined by a geodesic segment. 2. Triangle or Distance Comparison. Every geodesic triangle in X with perimeter < 2D(~:) is at least as thin as the corresponding triangle in M~. More precisely: for each geodesic triangle A in X with sides al, a~, % with Perimeter(A) ----- [ ~1 ] + I a~ I + [%]< 2D(~) we construct a REFERENCES 122 BRUCE KLEINER AND BERNHARD LEEB comparison triangle A in M~ with sides ~'~ satisfying ]~'~ I = ] e~ I. Each point x on A corresponds to a unique point ~ on A which divides the corresponding side in the same ratio. We require that for all x~, x~ e A we have I xx x~ [ ~< I ~'~ ~'2 [. Remark 9.. 1.1. -- Note that we do not require X to be locally compact. Also, X need not be path connected when K > O. This is slightly more general than some other definitions in the literature. Example 2.1.2. -- A complete 1-connected Riemannian manifold with sectional cur- vature <<. ~ <<. 0 and all its closed convex subsets are CAT(~)-spaces. In particular, Hadamard manifolds are CAT(0)-spaces. This is why we will also call CAT(0)-spaces Hadamard spaces. Example 9..1.8 (Berestovski). -- Any simplicial complex admits a piecewise spherical CAT (1) metric. Condition 2 implies that any two points xl, x~ with [ x 1 x~ ] < D(~:) are connected by precisely one geodesic; hence we may speak unambiguously of x 1 x, as the geodesic segment joining x 1 to x,. The CAT(~:)-spaces for ~: ~< 0 are contractible geodesic spaces. To see that upper curvature bounds behave well under limiting operations, it is convenient to use an equivalent definition of CAT(n)-spaces which only refers to finite configurations of points rather than geodesic triangles. If v, x, y, p e X, and 3', ~, 3, P e M~ we say that 3', ~', y, ~ form a ~-comparison quadruple if 1. ~" lies on ~,~'. 2. llvxl-I l[< , llvyl-I' Yll< , Ilxyl-l' Ylt< , Ilxpl-I ll< , IIpyl - I Yll < By a compactness argument, we note that there exists a function ~(P, a)> 0 such that for every ~ > 0, and every quadruple of points v, x, y, p in a CAT()-space X satisfying I vx I + [ xy [ § lyv ] < P < 2D(~:), each ~(P, ~)-comparison quadruple v_ ~,..~, ~ satisfies I vp I<~ + ~. We will refer to this condition as the ~-four-point condition. It is a closed condition on four point metric spaces with respect to the Hausdorff topology. A complete metric space X is a CAT(~:)-space if and only if it satisfies the ~-four-point condition and every pair of points x, y e X with I xy I < D (~:) has approximate midpoints, i.e. for every s' > 0 there is a m e X with [ xm [, ] my ] < [ xy 1/2 + a'. To see this, note that in the presence of the ~-four-point condition approximate midpoints are close to one another, so one may produce a genuine midpoint by taking limits. By taking successive midpoints, one can produce a geodesic segment. 2.1.2. Coning Let Z be a metric space with Diam(Z) ~< ~. The metric cone C(E) over Y~ is defined as follows. The underlying set will be Z � [0, oo)/~ where ~ collapses X � { 0 } to RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 123 a point. Given vl, v2 9 Z, we consider embeddings p : { vl, v2 } � [0, 00) ~ E 2 such that [ O(v~, t) = ] t I and/o(p(v~, t~), O(v2, t2)) = [ v~ v~ ], and we equip C(Z) with the unique metric for which these embeddings are isometric. The space C(E) is CAT(0) if and only if E is CAT(l). 2.1.3. Angles and the space of directions of a CAT(K)-space Henceforth we will say that a triple v, x, y defines a triangle A(v, x.y) provided l vx [ + [xy[ + iyv l< 2 Diam(M~). The symbol 7(x,y) will denote the angle of the comparison triangle at the vertex ~'. If x', y' are interior points on the segments vx, vy, then "~,(x',y')<~ "fi~(x,y). From this monotonicity it follows that lim,,,u, ~ , ~,(x',y') exists, and we denote it by /~(x,y). This definition of angle coincides with the notion of the angle between two segments in the Riemannian case. One checks that one obtains the same limit if only one of the points x', y' approaches v: (3) /_.(x,y) = lim /_~(x',y); /~ satisfies the triangle inequality. Note that from the definition we have (4) /~(x,y) <~ "~,(x,y). In the equality case a basic rigidity phenomenon occurs: Triangle Filling Lemma 2.1.4. -- Let x, y, v be as before. If /_~(x,y) = 7 (x,y), then also ~he other angles of the triangle A(v, x,y) coincide with the corresponding comparison angles; moreover the region in M~ bounded by the comparison triangle can be isometrically embedded into X so that corresponding vertices are identified. The angles of a triangle depend upper-semicontinuously on the vertices: Lemma 2.1.5. -- Suppose v, x,y 9 X define a triangle, v ~t" x,y, and v k ~ v, x k -+ x, y~ -+y. Then v~, Xk, y, define a triangle for almost all k and lim sup Z_,k(xk,y~) ~ /~(x,y). In the special ease that vk 9 vx~ -- { v } holds lim~_~ | /,k (xk, yk) =/__,(x,y) and lim~_~ Z.k(v,y~) = 7: --/_.(x.y). Pro@ -- For x' e ~ -- { v } and y' 9 ~ -- { v } we can choose sequences of points -- i t ~ N ! t x~ 9 v k xk,y ~ 9 vky k with x k --> x' and Yk -+Y'- Then /__.,(xk,y,) ~ /--,k(x~,,y'~) -->/__.(x ,y ) and the first assertion follows by letting x',y'~v. If v,e-~--{v.x,} then /_.(x,,y,) <~ anglesum(A(v, v,,y,)) -- /~k(v,y,) and 7: -- /_,k(v,y,) ~ 7--~k(x~,y,) while lira sup anglesum(A(v, vk.yk) ) ~< re. Sending k to infinity, we get /_,(x,y) ~ rc -- lim inf/.k(v, yk)~< lim inf Z.k(x,,y,) and hence the second assertion. 124 BRUCE KLEINER AND BERNHARD LEEB The condition that two geodesic segments with initial point v e X have angle zero at v is an equivalence relation; we denote the set of equivalence classes by Y.* X. The angle defines a metric on ~; X, and we let Z~ X be the completion of Z; X with respect to this metric. We call elements of Y.. X directions at v (or simply directions), and ~ denotes the direction represented by ~. We define the logarithm map as the map log. = logzox:B.(D(K))\v ~X. X which carries x to the direction ~. The tangent cone of X at v, denoted C. X, is the metric cone C(Z. X); we have a logarithm map log. =logc. x : B.(D(~:)) ~ C. X. Given a basepoint v ~ X, x ~ X with d(v, x) < D(~:), and X e [0, 1], let Xx e X be the point on v-'-x satisfying [v(Xx)[/] vx [= X. We define a family of pseudo- metrics on B.(D(~:)) by d.(x,y)= d(zx, ~y)/s. They converge to a limit pseudo- metric do. The pseudo-metric space (B.(D(K)), d.) satisfies the ~-four-point condition, so the limit pseudo-metric space (B.(D(~:)), do) satisfies the ~0-four-point condition. But do(x,y) = d(log, x, log.y) where log. : B.(D(K)) ~ C. X is the logarithm defined above, so we see that the tangent cone C. X satisfies the ~0-four-point condition (C(Z; X) is dense in C. X, and every four-tuple in C(Y,; X) is homothetic to a four-tuple in log.(Bo(D(~:))). If zx is the midpoint of the segment (Zx) (Xy), then d(log~ x, log.y) = lim 1 d(~, ~y) r 0 = lim 2 d(~x, z~) = lim 2 d(z., ~y) e--~0 ~ e-+0 >/ max 2d log~ x, - 1 log~ z. , !im ~ 2d log. x, - log~ z, . So C, X also has approximate midpoints. Since C, X is complete, it is a CAT(0)-space; consequently X X is a CAT(1)-space. This fact is due to Nikolaev [Nik]. 2.2. CAT(1)-spaces CAT(1)-spaces are of special importance to us, because they will turn up as spaces of directions and Tits boundaries of Hadamard spaces. 2.2.1. Spherical join Let B1 and B~ be CAT(1)-spaces with diameter Diam(B,)~< n. Their spherical join B 1 o B2 is defined as follows. The underlying set will be B 1 � [0, ~/2] � B~/~, where " ~-~ " collapses the subsets { b 1 } � { 0 } � B~ and B x � { ~/2 } � { b~ } to points. Given b,,b~ ~B, (i= 1,2), we consider embeddings p:{bl, b' 1) � [0, n/2] � s. We think ofS s as the unit sphere in C z and require that t ~ ?(bl, t, bz) and t' v-~ p(b~, f, b'~) are unit speed geodesic segments whose initial (resp. end) points lie on the great circle S 1 � { 0 } (resp. { 0 } � S 1) and have distance d~(bl, b'l) (resp. dB2(bz, b~)). The distance RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 125 of the points in B 1 o B 2 represented by (bl, t, b~) and (b'~, t', b~) is then defined as the (spherical) distance of their p-images in $3; it is independent of the choice of p. To see that B 1 o B 2 is again a CAT(1)-space and that the spherical join operation is associative, observe that the metric cone C(B1 o Be) is canonically isometric to C(B1) � C(B2) and that the product of CAT(0)-spaces is CAT(0). The metric suspemion of a CAT(1)-space with diameter ~< ~ is defined as its spherical join with the CAT(1)-space { south, north } consisting of two points with distance ~. Lemma 2.2.1. -- Let B 1 and B, be CAT(1)-spaces with diameter ~ and suppose s is an isometrically embedded unit sphere in the spherical join B = B 1 o B2. Then there are isometrically embedded unit spheres s, in B~ so that sl o s2 contains s. Proof. -- We apply lemma 2.3.8 to the metric cone C(B) ~ C(B1) � C(B~). The set C(s) is a flat in C(B) and hence contained in the product of flats F, _c C(Bi) ' h := 0Tim F, is a unit sphere in B~ and s 1 o s~ = 0Tlt.(F 1 X F2) ~- 0~l~ C(s) : s. [] 2.2.2. Convex subsets and their poles We call a subset C of a CAT(1)-space B convex if and only if for any two points p, q ~ C of distance d(p, q) < n the unique geodesic segment p-q is contained in C. Closed convex subsets of B are CAT(1)-spaces with respect to the subspace metric inherited from B. Basic examples of convex subsets are tubular neighborhoods with radius ~< 7~/2 of convex subsets, e.g. balls of radius ~< n/2. Suppose that C C B is a closed convex subset with radius Rad(C) >/ ~, i.e. for each p e C exists q ~ C with d(p, q) >i ~. We define the set of poles for C as Poles(C) := {, eB: d(,, .)It- 2}. IfDiam(C) > r~ then C has no pole. If Diam(C) ---- Rad(C) = r~ then Poles(C) is closed and convex, because it can be written as an intersection Poles(C) = ['lr of convex balls. The convex hull of C and Poles(C) is the union of all segments joining points in C to points in Poles(C), and is canonically isometric to C o Poles(C). This follows, for instance, when one applies the discussion in section 2.3.3 to the parallel sets of C(C) in the metric cone C(B). Consider the special case that C consists of two antipodes, i.e. points with distance ~, 4 and ~. Then the convex hull of { 4, ~ } and Poles({ 4, ~ }) is exactly the union of mini- mizing geodesic segments connecting 4, ~ and it is canonically isometric to the metric suspension of Poles({ 4, ~ }). 2.3. Hadamard spaces We will call CAT(0)-spaces also Hadamard spaces, because they are the synthetic analog of (closed convex subsets in) Hadamard manifolds, i.e. simply connected complete manifolds of nonpositive curvature, cf. example 2.1.3. 126 BRUCE KLEINER AND BERNHARD LEEB 2.3.1. The geometric boundary Let X be a Hadamard space. Two geodesic rays are asymptotic if they remain at bounded distance from one another, i.e. if their Hausdorff distance is finite. Asympto- ticity is an equivalence relation, and we let 0~o X be the set of equivalence classes of asymptotic rays; we sometimes refer to elements of O~ X as ideal points or ideal boundary points. For any point x ~ X and any ideal boundary point ~ e 0~ X there exists a unique ray x~ starting at x which represents ~. The pointed Hausdorff topology on rays emanating from x ~ X induces a topology on ~ X. This topology does not depend on the base point x and is called the cone topology on 0oo X; Oo~ X with the cone topology is called the geometric boundary. The cone topology naturally extends to X u 0~o X. If X is locally compact, then O~ X and X := X w ~o X are compact and X is called the geometric compactification of X. 9,.8.2. The Tits metric Earlier we defined the angle between two geodesics vx, vy at v e X by using the monotonicity of comparison angles ~(x',y') as x' --+ v, y' --+ v. Now we consider a pair of rays v~, v--~, and define their Tits angle (or angle at infinity) by (5) Lxit,(~ , ~) := lim L~(x',y') 0Y--~ ~, ~' --~ ~ where x' e v~ and y' e ~-~. LTlt~ defines a metric on 0r X which is independent of the basepoint v chosen. We call the metric space 0~it~ X := (0~o X,/-~Lt~) the Tits boundary of X and L~t~ the Tits (angle) metric. The estimate L~(x ,y ) = r: -- L,,(v,y') -- ~r x') <~ L,,(~,y') ~'-~ >0 y' "-*"4 implies, combined with (4): N # t /~(~, .4) <<. L,(x ,y ) <<. L,.(~, .4). Consequently, the Tits angle can be expressed as (6) L:r~(~, ~) = lim L,,,~(~, ~) for any geodesic ray r:R + -+ X asymptotic to ~ or ~, and also as: (7) LT,t.(~ , ~) = sup L,(~, ~). z~X RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 127 Still another possibility (the last one which we will state) to define the Tits angle is as follows: If r~:R + ~ X are geodesic rays asymptotic to ~, then (8) 2 sin/~t~(~l, ~) lim d(q(t), r2(t)) 2 ~-~| t The next lemma relates the cone topology on 0oo X to the Tits topology. Fix v e X and consider the comparison angle 7_o : (x\{ ~ }) x (x\{ ~ }) -+ [0, =]. By monotonicity, it can be extended to a function ~_~ : (X\{ ~ }) x (X\{ ~ }) --. [0. =]. Note that for ~, ~ ~ O~ X, we have ~,(~, ~l) = Lmtt~(~, ~). Lemma 2.3.1 (Semicontinuity of comparison angle). -- The angle ~ is lower semi- continuous with respect to the cone topology: If xk,y~, ~, ~ e X -- { v } are such that ~ ---- timk_~ o x k and ~ = limk_~=yk, then ~(~, ~) <<. lim inf~_~ ~ ~_,(x~,y~). Proof. -- We treat the case 4, ~ ~ O~ X, the other cases are similar or easier. Since the segments (or rays) vxk, vy k are converging to the rays v~, v-~ respectively, we may l t t ! choose x; e ~ andy;~ e v~, such that I xk v ], IY, v [ -+ oo and d(x,, v~) -+ O, d(y,, -~) -+ O. Hence by triangle comparison we have Z,(x~,y~) >I/_,(x~,y~) ~ ' ' -+ A~,~(~,~). [] Lemma 2.3.2. ~ Every pair ~, ~ E O~ X with /Ti~(~, ~) < 7: has a midpoint. Proof. -- Pick v E X. Take sequences x~ ~ v~, y, ~ v-~ with I x~ [ ----- [y~ [ -+ m. Let m~ be the midpoint of x--~. Since A(v, x~,y,) is isosceles, "~,(x~, m,) = ~,(m~,y~) <~ 7,(x,,y,)/2, by lemma 2.3.1 it suffices to show that b~ converges to a ray v-~, for some ~ ~ O~ X. For i<j, set ~s := I vx~ [/[ vxj [. By triangle comparison, we have the following inequalities: 1 x~(x. m,) I -< x~, t *~ m, i X. = -ff [ x,y, I. ~k~j [Y,(X~s ms) I -< x.s l Ys m, I = -~- I xjYs 1, I ..(N, m,) I + ly.(x., m,) ] ~> I Y. t- Since Ns. (1 xjyj l/I am I) -~ 1 as i,j -+ oo, we have [ mi(X,s mj) I § 0 Ix,(N~ms) I ~-1, ly~(Nsms) I ~-1 => Ix, m,l Imm, I I x~ m. I 128 BRUCE KLEINER AND BERNHARD LEEB and, since L~(~, ~) < ~, this in turn implies: ] mi(X~j mj) [ § 0. Ivmi[ Fixing t > 0, if we set t/[ vm, I : ~,, then [ (~1, m,) (11, X,~ m~) I -+ 0 as i,j -+ oo. Since [ v(~l~ m~)[= t, this shows that the segments vm---~ converge in the pointed Hausdorff topology to a ray v--~ as desired. [] The completeness of X implies that (0~ X,/~lt.) is complete. The metric cone C(0~o X,/Tit~) (the Tits cone) is complete and has midpoints. Moreover, since every quadruple in C(0oo X, /~t~) is approximated metrically (up to rescaling) by quadruples in X, C(0oo X,/-T~t~) satisfies the 80-four-point condition and is therefore a CAT(0)-space. By section 2.1.1 we conclude: Proposition 2.3.3. -- The Tits boundary of a Hadamard space is a CAT(1)-space. There is a natural 1-Lipschitz exponential map exp~ : C(0~lt, X) -+ X defined as follows: For [(4, t)] ~ C(0T~t~ X) = 0T~ ~ X � [0, oo)/~, let exp~[(~, t)] be the point on p~ at distance t from p. The logarithm map log~ : X -- { p } -+ Z r X extends continuously to the geometric boundary and induces there a 1-Lipschitz map log~ : 0T~t~ X -+ Y,~ X. The Triangle Filling l.emma 2.1.4 implies the following rigidity statement: Flat Sector Lemma 23.4. -- Suppose the restriction of log~ :OTt ~X~ZvX to the subset A ~ 0Tit~X is distance-preserving. Then the restriction of exp~ : C(OTit~ X) -+ X to C(A) _c C(0Tit, X) is an isometric embedding. 2.3.3. Convex subsets and parallel sets A subset of a Hadamard space is convex if, with any two points, it contains the unique geodesic segment connecting them. Closed convex subsets of Hadamard spaces are Hadamard themselves with respect to the subspace metric. Important examples of convex sets are tubular neighborhoods of convex sets and horoballs. We will denote by HB~(x) the horoball centered at the point ~ ~ O~ X and containing x e X in its boundary. Let C1 and C, be closed convex subsets of a Hadamard space X. Then by (4), the distance function d(., C2)ICl = dc,]~ : C1 -+ R~>o is convex and the nearest point projection nc,[c~ :C1-+ C~ is distance-nonincreasing; dc, le x is constant if and only if ne,,]c~ is an isometric embedding. In this situation, we have the following rigidity statement: Flat Strip Lemma 2 3.5. -- Let C 1 and C~ be closed convex subsets in the Hadamard space X. If de, lo x - d then there exists an isometric embedding + : C 1 x [0, d] ~ X such that +(., 0) = idet and +(., d) = o,]o1" RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 129 This is easily derived from the Triangle Filling Lemma 2.1.4, respectively from the following direct consequence of it: Flat Rectangle Lemma 2.3.6. -- Let x~ e X, i e Z/4Z, be points so that for all i holds /_zi(x,_l, xi+~)>~ 7~/2. Then there exists an embedding of the fiat rectangular region [0, I xo Xx I] � [0, I xl x21] C E' into X carrying the vertices to the points x,. We call the closed convex sets Cx, Cz -- X parallel, Ca I I c~, if and only if dct It, and dca[c 2 are constant, or equivalently, 7~ealc a and ~cxle, are isometries inverse to each other. Being parallel is no equivalence relation for arbitrary closed convex subsets. However, it is an equivalence relation for closed convex sets with extendible geodesics, because two such subsets are parallel if and only if they have finite Hausdorff distance. (A Hadamard space is said to have extendible geodesics if each geodesic segment is contained in a complete geodesic.) Let Y _~ X be a closed convex subset with extendible geodesics. Then Rad(0a~ ~ Y) = 7:. The parallel set Py of Y is defined as the union of all convex subsets parallel to Y; Py is closed, convex and splits canonically as a metric product (9) Pr ~ Y x Ny. Here Ny is a Hadamard space (not necessarily with extendible geodesics) and the subsets Y � { pt } are the convex subsets parallel to Y. The cross-sections of Py orthogonal to these convex subsets can be constructed as intersections of horoballs: (10) {y} x Ny=Py n ['1 HB~(y) VyeY. ~ 0Tits Y Applying the Flat Sector Lemma 2.3.4 one sees furthermore that 0Ti ~ Ny is canonically identified with Poles(OT, t, Y) C 0Tj ~ X; 0Tit, Py is the convex hull in 0T,t, X of OTi ~ Y and Poles(0T[ ~ Y) and we have the canonical decomposition: (1l) 0Tlta Py ~ � Y o Poles(0Tl ~ Y). 2.3.4. Products The metric product of Hadamart spaces X~ is defined as usual using the Pythegorean law. It is again Hadamard and its Tits boundary and spaces of directions decompose canonically: (12) x ... x x.) = OT, X, o ... o X., (is) X m ..... ~,~(Xl X ... X X,) =Z~IX lo...oZ,.x.. Proposition 2.3.7. -- If X is a Hadamard space with extendible geodesics then all join decompositions of Or~t, X are induced by product decompositions of X. 17 130 BRUCE KLEINER AND BERNHARD LEEB Proof. -- Assume that the Tits boundary decomposes as a spherical join 0a~ ~X--B~oB_I and consider, for xeX and i=+ 1, the convex subsets C~(x) := [']~e~_iHB~(x) obtained from intersecting horoballs. Using extendibility of geodesics, i.e. Rad Y., X = ~, one verifies that 0~, C~ ---- B~, C~ has extendible geodesics and C a(x) are orthogonal in the sense that Z, C,(x) = Poles(Y., C ,(x)). Furthermore any two sets Cx(x) and C_a(x') intersect in the point ~cl~,l(x') -~ ~e_~(x). The assertion follows by applying the Flat Rectangle Lemma 2.3.6. [] Lemma 2.3.8. -- Let X 1 and X 2 be Hadamard spaces and suppose that F is a flat in the product space X = X 1 � X 2. Then there are flats F~ ~_ X~ so that F 1 � F2 ~ F. Proof. -- Consider unit speed parametrizations c, c':R ~ F for two parallel geodesics y, y' in F. Then c~ := nxi o c and c~ := ~xi o c' are constant speed parame- trizations for geodesics y~, 7~ in X i. Since the distance functions d := dx(c , c') and d~ := dxi(C~, c~) are convex, satisfy d ~ ~-- d~ + d~ and d is constant, it follows that the d~ are constant, i.e. y~ and y~ are parallel. Since this works for any pair of parallel geodesics contained in F, it follows that ~xi F is a flat in F,. [] 2.3.5. Induced isomorphlsms of Tits boundaries We now show that any (1, A)-quasi-isometric embedding of one Hadamard space into another induces a well-defined topological embedding of geometric boundaries which preserves the Tits distance. Proposition 2.3.9. -- Let X 1 and X z be Hadamard spaces and suppose that 9 : X 1 ~ X z is a (1, A)-quasi-isometric embedding. Then there is a unique extension ~:X l -+ X2 such that 1. ~(9~ X1) ~ ~ X,, 2. ~ is continuous at boundary points, 3. ~]0~xl is a topological embedding which preserves the Tits distance. We let 0~o ~ d,~ ~ [e~o x" Proof. -- We first observe that there is a function r (depending on A but not on the spaces X 1 and X2) with r as R-+oo such that ifp, x, yeX t and d(p, x), d(p,y) > R then (14) [ 2,(x,y) -- ~_~,,,(~(x), ~(y) I < ~(R)). Lemma 2.3.10. -- Suppose that x~ is a sequence of points in X 1 which converges to a boundary point 41. Then ~(x~) e X z converges to a boundary point 42. Proof of Lemma. -- Pick a base pointp. There are points y~ ~ ~ such that d (p, y,) -+ oo and lin~.i_~ ~ "~(y~,yj) = 0. By (14), the points ~(y~) converge to a boundary point ~2. Applying (14) again, we conclude that ~(x,) converges to 42 as well. [] RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 131 Proof of Proposition continued. -- From the previous lemma we see that if x~ and x~ are sequences in X1 converging to the same point in 0o~ X1 then the sequences ~(x~) and tl)(x~) converge to the same point in 0~ X 2. This allows us to extend 9 to a well- defined map ~ : X x ~ 2 2. We now prove that ~ is continuous at every boundary point ~. Let x~ e X~ be a sequence of points converging to ~ e 0~ X 1. By the lemma, we may choose y~ e Xx withy~ epx---~ so that for every R the Hausdorff distance between ~(p) r n BR(~(p)) and @(p) ~(xi) n BR(~(p)) tends to zero as R -+ oo. Hence lim~_~ ~(x~) = lim~_~ ~(y~) = ~(~) by the lemma. Another consequence of the lemma is that the image ray q)(p~) diverges sublinearily from the ray ~(p) ~(~) in the sense that lim 1 .dn(C(p-~ c~ B~(p)), r ~(~) n B~(C(p))) = 0 where d~ denotes the Hausdorff distance. This implies that 0| r a~ ~]eoox~ preserves the Tits distance and is a homeomorphism onto its image. [] 2.4. Ultralimits and asymptotic cones The presentation here is a slight modification of [Gro], see also [KaLe]. 2.4.1. Ultrafilters and ultrallmlts Definition 2.4.1. -- A nonprincipal ultrafilter is a finitely additive probability measure co on the subsets of the ~mtural numbers N such that 1. co(S) = 0 or 1 for every S C N. 2. co(S) = 0 for every finite subset S C N. Given a compact metric space X and a map a : N -+ X, there is a unique element co-lim a e X such that for every neighborhood U of c0-1im a, a- 1 (U) C N has full measure. In particular, given any bounded sequence a : N -+ R, co-lim a (or %) is a limit point selected by co. 2.4.2. Ultralimlts of sequences of pointed metric spaces Let (X~, di, ,i) be a sequence of metric spaces with basepoints ,~. Consider X~ = { x EII, e s X~ ] d,(x,, ,,) is bounded }. Since d,(x,,y,) is a bounded sequence we may define d'~: X~o x X~ ~R by d"~(x,2)= r '~ is a pseudo- distance. We define the ultralimit of the sequence (X~, d~, "i) to be the quotient metric space (X~,d~), x,o e X~ denotes the element corresponding to x = (xi)eX~. *~ := (,~) is the basepoint of (X~, d~). 132 BRUCE KLEINER AND BERNHARD LEEB Lemma 2.4.2.- If (X,, d,, ,,) is a sequence of pointed metric spaces, then (Xo, do, *o) is complete. Proof. -- Let x,~ be a Cauchy sequence in X~,, where Xo ~ = e0-1im x~. Let Nx = N. Inductively, there is an o-full measure subset Ni ~_ Nj_ x such that i ~ Nj implies [d~(~,x~) --do(x ~,x') 1< 1/2 ~, for 1~< k, l.<<j. For ieN~--N~_l, define y,=xl. Then x,~ -+y~. [] The concept of ultralimits is an extension of Hausdorff limits. Lemma 2.4.8. -- If (X,, d,, ,,) form a Hausdorff precompact family of pointed metric spaces, then (Xo, do, *o) is a limit point of the sequence (X~, d~, *~) with respect to the pointed Hausdorff topology. Proof. -- To see this, pick c, R, and note that there is an N such that we can find j N i for an N element sequence { x~ }i = 1 C X~ which is c-dense in X~. The N sequences x~ l~<j~< N give us N elements in x~X o. Ifyo eXo,yoeB*o(R), then for ~-a.e. (that is, u-almost every) i, di(y~,,~) < R. Consider do(y~, , xo). J Given g~> 0, [ d,~(yo, x~) -- d~(y,, xr [ < g for ~-a.e. i, which implies that d,~ (y,~, x~) < ~ for some 1 ~< j ~< N. Hence we have seen that B.,~(R) is totally bounded, and for all g > 0 there is an t-net in B.o(R ) which is a Hausdorff limit point of c-nets in the X~'s. It follows that (X~, d~, *i) subconverges to (Xo, d,~, ,~,) in the pointed Hausdorfftopology. [] In general, the ultralimit Xo, is not Hausdorffclose to the spaces X~ in the " approxi- mating " sequence. However, the Hausdorff limits of any precompact sequence of subspaces Yi C X~ canonically embed into X~. The importance of ultralimits for the study of the large-scale geometry derives from the following fact: If for each i, f~:X~-+ Y~ is a (L, C)-quasi-isometry with d~(f(,~), *~) bounded, then the f induce an (L, C)-quasi-isometryf~ : X o -+ Y,~. It follows that if for each i, and every pair of points a~, b~ e X~ the distance d~(a~, b~) is the infimum of lengths of paths joining a~ to b~ then every pair of points a~, bo ~ X,~ is joined by a geodesic segment. Lemma 2.4.4. -- If (X,, d~, **) is a CAT(~:)-space for each i, then so is (Xo, do, *o). If do(ao, b~,) < D(t:), then the geodesic segment ao b o is an ultralimit of geodesic segments. If <~ 0 and each X~ has extendible geodesics then each ray (respectively complete geodesic) in X,~ is an ultralimit of rays (respectively complete geodesics) in the X/s. Proof. -- If each (X~, d~, ,~) is a CAT(t:) length space, then clearly (X,~, do, *o) satisfies the 8.-four-point condition since this is a closed condition. Hence (Xo, d~, ,~) is a CAT(n) length space since it is a geodesic space satisfying the 8,-four-point condition. If ao, b~ ~Xo with l ao b,0 [ < D(,:), then there is a unique geodesic segment joining ao to b o. On the other hand, if a o ---- c0-1im a~, b o = c0-1im b~, then the ultra- limit of the geodesic segments a, b, is such a geodesic segment. RIGIDITY OF O).UASI-ISOMETRIES FOR SYMMETKIC SPACES AND EUCLIDEAN BUILDINGS 133 Now suppose a,~,~ a~,' . .. determine a ray, in the sense that d~(a'~, ~) = do(a'~, a~) + a~(<, ~) for i~< j ~ k. Let N 1 = N. Inductively, there is an (~-full measure N~ ___ N~_, such that a ~ a~ is within a 1/2~-neighborhood of the segment a ~ a~ for i e Nj, 0 ~< 1 ~< j. For i e N~ -- N i_ , extend the segment a ~ a~ to a ray a~ ~ ~ with initial point a ~ Then the ultralimit of the sequence a ~ ~ is the ray we started with. The case of complete geodesics follows from similar reasoning. [] Lemma 2.4.5. -- Suppose that there is a D > 0 such that for each i, Isom(X~.) has an orbit which is D-dense in X~. If ~ > 0 and ),~ --+ O, then the ultralimit of (X~., ~. d~, *~) is independent of the choice of basepoints ,~, and has a transitive isometry group. 2.4.3. Asymptotic cones Let X be a metric space and let ,~ ~ X be a sequence of basepoints. We define the asymptotic cone Cone(X) of X with respect to the non-principal ultrafilter % the sequence of scale factors Z, with r k, = oo and basepoints *,, as the ultralimit of the sequence of rescaled spaces (X,, d,, *,) := (X, Z~-l.d, *,). When the sequence 9 .-= * is constant, then Cone(X) does not depend on the basepoint , and has a canonical basepoint **~ which is represented by any sequence (x,)C X satisfying r k~ -1 .d(x., ,) = 0, for instance, by any constant sequence (x). Proposition 2.4.6. -- 9 If X is a geodesic metric space, then Cone(X) is a geodesic metric space. 9 If X is a Hadamard space, then Cone(X) is a Hadamard space. 9 IfX is a CAT(,c)-spacefor some ~ < O, then Cone(X) is a metric tree. 9 If the orbits of Isom(X) are at bounded Hausdorff distance from X, then Cone(X) is a homogeneous metric space. 9 A (L, C) quasi-isometry of metric spaces ?:X-+Y induces a bilipschitz map Cone(~) : Cone(X) -+ Cone(Y) of asymptotic cones. Assume now that X is a Hadamard space. Let (F,,),E~ be a sequence of k-flats in X and suppose that ~-lim, k~-l.d(F,, ,) < ~. Then the ultralimit of the embeddings of pointed metric spaces m ll k is a k-flat R ,-+ Cone(X) in the asymptotic cone. We denote the family of all k-flats in Cone(X) arising in this way by ~'(k). 134 BRUCE KLEINER AND BERNHARD LEEB 3. SPHERICAL BUILDINGS Our viewpoint on spherical buildings is slightly different from the standard one: for us a spherical building is a CAT(l) space equipped with an extra structure. This viewpoint is well adapted to the needs of this paper, because the spherical buildings which we consider arise as Tits boundaries and spaces of directions of Hadamard spaces. Apart from the choice of definitions and the viewpoint, this section does not contain anything new; the same results and many more can be found (often in slightly different form) in [Til, Ron, Brbk, Brnl, Brn2]. 3.1. Spherical Coxeter complexes Let S be a Euclidean unit sphere. By a reflection on S we mean an involutive isometry whose fixed point set, its wall, is a subsphere of codimension one. If W C Isom(S) is a finite subgroup generated by reflections, we call the pair (S, W) a spherical Coxeter complex and W its Weyl group. The finite collection of walls belonging to reflections in W divide S into isometric open convex sets. The closure of any of these sets is called a chamber, and is a funda- mental domain for the action of W. Chambers are convex spherical polyhedra, i.e. finite intersections of hemispheres. A face of a chamber is an intersection of the chamber with some walls. A face (resp. open face) of S is a face (resp. open face) of a chamber of S. Two faces of S are opposite or antipodal if they are exchanged by the canonical involution of S ; two faces are opposite if and only if they contain a pair of antipodal points in their interiors. A panel is a codimension 1 face, a singular sphere is an intersection of walls, a half-apartment or root is a hemisphere bounded by a wall and a regular point in S is an interior point of a chamber. The regular points form a dense subset. The orbit space Amo a := S/W with the orbital distance metric is a spherical polyhedron isometric to each chamber. The quotient map (15) 0=0 s:S ~Amo a is 1-Lipschitz and its restriction to each chamber is distance-preserving. For 8, 8' c Amo a, we set D(8, 8'):= { ds(x , x')lx , x' cS, 0x = 8, 0x' = 8') and 9+(8) := D(8, 8)\{ 0 }. Note that D + is continuous on each open face of Amo a. An isomorphism of spherical Coxeter complexes (S, W), (S', W') is an isometry : S -~ S' carrying W to W'. We have an exact sequence 1 ~ W ~ Aut(S, W) ~ Isom(A~oa) ~ 1. RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 135 Lemma 3.1.1. ~ If g ~ W, then Fix(g) ~_ S is a singular sphere. If Z C S then the subgroup of W fixing Z pointwise is generated by the reflections in W which fix Z pointwise. Proof. -- Every W-orbit intersects each closed chamber precisely once. Therefore the stabiliser of a face ~ C S fixes ~ pointwise. So for all g 6 W, Fix(g) is a subsphere and a subcomplex, i.e. it is a singular sphere. By the above, without loss of generality we may assume that Z is a singular sphere. Let W z be the group generated by reflections fixing Z pointwise. If ~ is a top-dimensional face of the singular sphere Z then each W-chamber containing c is contained in a unique Wz-chamber; therefore W z acts (simply) transitively on the W-chambers containing ~. Since W acts simply (transitively) on W-chambers, it follows that Fixator(Z) = Fixator(a) ----- W z. [] 3.2. Definition of spherical buildings Let (S, W) be a spherical Coxeter complex. A spherical building modelled on (S, W) is a CAT(1)-space B together with a collection d of isometric embeddings t:S---~ B, called charts, which satisfies properties SB1-2 described below and which is closed under precomposition with isometrics in W. An apartment in B is the image of a chart ~ : S ~ B; t is a chart of the apartment ~(S). The collection d is called the atlas of the spherical building. SB1. Plenty of apartments. -- Any two points in B are contained in a common apartment. Let ~A1, ~A, be charts for apartments A1, Az, and let C = A 1 n As, C' = t.Asl(C ) C S. The charts tAi are W-compatible if ~-A~I o ~As [c' is the restriction of an isometry in W. SBg. Compatible apartments. -- The charts are W-compatible. It will be a consequence of corollary 3.9.2 below that the arias ~/ is maximal among collections of charts satisfying axioms SB1 and SB2. We define walls, singular spheres, half-apartments, chambers, faces, antipodal points, antipodal faces, and regular points to be the images of corresponding objects in the spherical Coxeter complex. The building is called thick if each wall belongs to at least 3 half-apartments. The axioms yield a well-defined 1-Lipsctfitz anisotropy map (1) (16) 0 B : B -+ S/W =: Amoa satisfying the discreteness condition: (17) dB(Xl, x,) e D(0~(x,), 0B(x,) ) V x~, x, ~ B. If ~ : S -+ S is an automorphism of the spherical Coxeter complex, then we modify the atlas by precomposing with ~; the atlases obtained this way correspond to symmetries of Amoa. (1) The motivation for this terminology comes from the role 0B plays in the structure of symmetric spaces and Euclidean buildings. 136 BRUCE KLEINER AND BERNHARD LEEB If ~' is an atlas of charts ,: S' -+ B giving a (S', W') building structure on B, then this spherical building is equivalent to (B, .~/) if there is an isomorphism of spherical Coxeter complexes 0~ : (S', W') -+ (S, W) so that .~' = { t o ~ [ t e ~ }. If B and B' are spherical buildings modelled on a Coxeter complex (S, W), with atlases ~ and M', an isomorphism is an isometry ? : B --* B' such that the correspondence ~ 9 o t defines a bijection ~ ~ ~'. 3.3. Join products and decompositions Let Bi, i = 1,..., n, be spherical buildings modelled on spherical Coxeter complexes (S~, W,) with atlases d i and spherical model polyhedra A~o d. Then W:= W 1 � ... � W, acts canonically as a reflection group on the sphere S = $1 o 9 9 9 o S,. We call the Coxeter complex (S, W) the spherical join of the Coxeter complexes (S,, W~) and write (18) (S, W) = (S~, W~) o ... o (S,, W,). The model polyhedron Amo d of (S, W) decomposes canonically as 1 n (19) Amo d = Amo d o ... o Am~. The CAT(1)-space (20) B = B1 o ... o B, carries a natural spherical building structure modelled on (S, W). The charts t for its atlas d are the spherical joins ~ = tx o ... o % of charts ~ e d~. We call B equipped with this building structure the spherical (building)join of the buildings B~. Proposition 3.3.1. -- Let B be a spherical building modelled on the Coxeter complex (S, W) with atlas d and assume that there is a decomposition (19) of its model polyhedron. Then: 1. There is a decomposition (18) of (S, W) as a join of spherical Coxeter complexes so that s, = I(AL ). 2. There is a decomposition (20) of B as a join of spherical buildings so that B~ -= O~l(~od). Proof. -- 1. We identify Amo d with a W-chamber in S and define S~ to be the minimal geodesic subsphere containing A~o a. Then S~ ~_ Poles(Sj) for all i 4=j and hence S = S x o ... o S. for dimension reasons. Each wall containing a codimension-one face of A~o a is orthogonal to one of the spheres S~ and contains the others. Hence W ~ W 1 � ... � W, where W, is generated by the reflections in W at walls orthogonal to S~. The group W~ acts as a reflection group on S~ and the claim follows. 2. Since any two points in B are contained in an apartment, one sees by applying the first assertion that the B~ are convex subsets and B is canonically isometric to the RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUGLIDEAN BUILDINGS 137 join of CAT(1)-spaces B = Bx o ... o B,. The collection of charts L[Bi, t ~ d~, forms an atlas for a spherical building structure on B, and B is canonically isomorphic to the spherical building join of the B,. [] We call a spherical polyhedron irreducible flit is a spherical simplex with diameter < n/2 and dihedral angles ~< ~/2 or if it is a sphere or a point. Accordingly, we call a spherical Coxeter complex (x) or a spherical building irreducible if its model polyhedron is irreducible. The spherical model polyhedron ~od has dihedral angles ~< n/2. A polyhedron of this sort has a unique minimal decomposition as the spherical join (19) of irreducible spherical simplices (which may be single points) and, if non-empty, the unique maximal unit sphere contained in ~oa. By Proposition 3.3.1, (19) corresponds to unique minimal decompositions (18) of the Coxeter complex (S, W) as a join of Coxeter complexes and (20) of B as a spherical building join. We call these decompositions the de Rham decompositions of (S, W) and B. The sphere factor in (19) occurs if and only if the fixed point set of the Weyl group is non-empty. We call the corresponding factor in the de R.ham decomposition the spherical de Rham factor. If W acts without fixed point, then ~oa is a spherical simplex (~) and the collection of chambers in S and B give rise to simplicial complexes. Lemma 3.3.9.. __ Let (S, W) be an irreducible spherical Coxeter complex with non-trivial Weft group W. Then for each chamber o there is a wall which is disjoint from the closure ~. Proof. -- Let 7' be a wall and p ~ S be a point at maximal distance rq2 from C. Pick a chamber ~' containing p in its closure. Then ~' n ~' = 0, because Diam(C) < =[2 due to irreducibility. Since W acts transitively on chambers, the claim follows. [] Proposition 3.3.3. -- Assume that B 1 and B2 are CAT(1)-spaces and that their join B = B x o B 2 admits a spherical building structure. Then the B, inherit natural spherical building structures from B. In particular, the spherical building B cannot be thick irreducible with non-trivial Weyl group. Proof. -- Applying lemma 2.2.1 to apartments in B, we see that there exist dt, d~ e N so that every apartment A ___ B splits as A = A t o A s where A,. is a d,-dimensional unit sphere in B,. Fix a chart t 0 in the atlas ~r for the given spherical building structure on B. Denote by $2 the d2-sphere t o x B2 in the model Coxeter complex (S, W) and by S 1 := Poles(S~) the complementary dl-sphere. The subgroup W 1 _~ W generated by reflections at walls containing S 2 acts as a reflection group on $1. Consider all charts e ~ with ~lss = t01Ss" The collection d 1 of their restrictions ~Isl forms an atlas for a spherical building structure on B t with model Coxeter complex (St, Wt). (t) This definition is slightly different form the usual one, which corresponds to irreducibility of linear representatiom. (s) By [GrBc] [theorem 4.2.4], Amoa is a simplex if W acts fixed point freely. Observe that having distance 1r than ~r/2 is an equivalence relation on the vertices. This implies the decomposition (19). 18 138 BRUCE KLEINER AND BERNHARD LEEB If B is thick, then its chambers are precisely the (closures of the) connected components of the subset of manifold points. Hence the joins al o a 2 of chambers ~, C B, are contained in chambers of B. So the chambers of B have diameter >/ z~/2 and B cannot be irreducible with non-trivial Weyl group. [] 3.4. Polyhedral structure Let A' be a face of z~o d and let a : A' ~B be the chart for a face in B, i.e. an isometric embedding so that 0 B o ~----idla,. Sublemma 3.4.1. -- The image a(Int A') is an open subset of Ogl(A'). Proof. -- Let x be a point in a(Int A') and assume that there exists a sequence (x,) in 0~a(A')\a(Int(A')) which converges to x. There are points x', E Im(a) with 0B(X~) = 0~(X,). Since 0 B has Lipschitz constant 1 and ~r is distance-preserving, we have aB(x., x) 0.(x)) = dB(x;,, x) and by the triangle inequality 2.a.(x., D+(%(x.)). -+0 Since D + is continuous on Int A', the right-hand side has a positive limit: lim D+(0B(x,)) = D+(0B(X)) > 0, a contradiction. [] [,emma 3.4. ~. -- Any two faces of B with a common interior point coincide. Consequentgy, the intersection of faces in B is a face in B. Proof. -- To verify the first assertion, consider two face charts al, ~r2 : A' ~ B of the same type. By Sublemma 3.4.1, { 8 ~ A' I al(8) = a2(8) } n Int A' is an open subset of Int A'. It is also closed, and hence empty or all of Int A' if A' is connected. If A' is disconnected, it must be the maximal sphere factor of Z~o a and all apartment charts agree on A'. Hence ~1 la' = ~r~la" also in this case. The intersections of two faces is a union of faces by the above; since it is convex, it is a face. [] As a consequence, the collection of finite unions of faces of B is a lattice under the binary operations of union and intersection; we will denote this lattice by ~r In the case that the Weyl group acts without fixed point, the chambers of B are simplices, and Jg'B is the lattice of finite subcomplexes of a simplicial complex. In general the polyhedron of this simplicial complex is not homeomorphic to B since it has the weak topology. RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 139 3.5. Recognizing spherical buildings The following proposition gives an easily verified criterion for the existence of a spherical building structure on a CAT(1)-space. Proposition 3.5.1. -- Let (S, W) be a spherical Coxeter complex, and let B be a CAT(1)- space of diameter ~ equipped with a 1-Lipschitz anisotropy map Os as in (16) satisfying the discreteness condition (17). Suppose moreover that each point and each pair of antipodal regular points is contained in a subset isometr# to S. Then there is a unique atlas d of charts ~ : S --.'- B forming a spherical building structure on B modelled on (S, W), with associated anisotropy map 0 B . Proof. -- The discreteness condition (17) implies that, for any face A' of A,,od , the restriction of 0~ to 0~ l(Int A') is locally distance-preserving and distance-preserving on minimizing geodesic segments contained in 0~l(IntA'). Therefore, if A C B is a subset isometric to S, the restriction of 0 B to A ~g :-~ A c~ 0Bl(IntAmod) is locally isometric and the components of A ~* are open convex polyhedra which project via 0~ isometrically onto Int Amo a. (17) implies moreover that A ~g is dense in A. Hence A is tesselated by isometric copies of Amo a and there is an isometry t A with 0~ o t A = 0 s which is unique up to precomposition with elements in W. If A x and A 2 are subsets isometric to S, and tax , ~A~ : S -+ B are isometrics as above then A x A 2 is convex, and we see that tax and tAz are W-compatible. We now refer to the isometrics t A : S -+ B as charts and to their images as apartments. The collection d of all charts will be the atlas for our spherical building structure. Since any point lies in some apartment, it lies in particular in a face, i.e. in the image of an isometric embedding a : A' -+ B of a face A' ___ Amo d satisfying 0 B o ~ = id la'" Lemma 3.4.2 applies and tile faces fit together to form a polyhedral structure on B. The apartments are subcomplexes. It remains to verify that any two points with distance less than n lie in a common apartment. It suffices to check this for any regular points Xl, x2, since any point lies in a chamber and an apartment containing an interior point of a chamber contains tile whole chamber (lemma 3.4.2). There is an apartment A x containing Xl. Consider a minimizing geodesic c joining xx and x~. By sublemma 3.4.1, A x is a neighborhood ofx 1. Hence near its endpoint Xl, c is a geodesic in the sphere A 1. Since B is a CAT(1)- space, we can extend c beyond x 1 inside A 1 to a minimizing geodesic ~ of length joining x 2 through x x to a point ~72 ~ Ax. By our assumption, the points x2, -~2 are contained in an apartment A,, which contains all minimizing geodesics connecting x 2 and s because x, is regular. In particular c- and therefore both points Xl, x~ lie in Aa. [] From the proof of Proposition 3.5.1 we have: Corollary 3.5.2. ~ Let B be a spherical building of dimension d, and let T c_ B be a subset isometric to the Euclidean unit sphere of dimension d. Then T is an apartment in B. RIGIDITY OF Q UASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 141 Proposition 3.6.4. -- The space Z~B together with the collection of embeddings Z~o ~A : Y,~, S ~ Y~ B as above is a spherical building modelled on (Y'~0 S, W~). If ~ ~ B is an antipode orB, then we have a 1-1 correspondence between apartments (respectively half-apartments) in B containing {p,~ } and apartments (respectively half-apartments) in ~-,~ B; Y~ B /s thick provided B is thick. Proof. -- Any two points pq~, pq, ~ Z~ B lie in an apartment; namely choose qx, q2 close to p, then any apartment A containing ql, qe will contain p and p~, E Y,~ A. So SB1 holds. The space Z~ B satisfies SB2 since we are only using charts tA : S -+ B with tA(P0) = P and B itself satisfies SB2. The remaining assertions follow immediately from the definition of the spherical building structure on Y~ B. [] 3.7, Reducing to a thick building structure A reduction of the spherical building structure on B consists of a reflection subgroup W'C W and a subset ~r ~' which defines a spherical building structure modelled on (S, W'). The Amod-direction map 0 3 can then be factored as rc o 0~ where 0~ : B -~ W'\S =: a~o d is the A~oa-direction map for the building modelled on (S, W'), and : w'\s -- A% = W\S is the canonical surjection. Proposition 3."/.1. -- Let B be a spherical building modelled on the spherical Coxeter complex (S, W), with anisotropy polyhedron Amo d = W~S. Then there exists a reduction (W', ~') which is a thick building structure on B; W' is unique up to conjugacy in W and ~' is determined by W'. In particular, the thick reduction is unique up to equivalence, so the polyhedral structure is defined by the CAT(l) space itself. The proof will occupy the remainder of this paper. We set d =-dim(B), Rs ={p e B]Z~ B is isometric to a standard Sa-1}, and S~ = B\R s. If p E B and 0 ~ 0 is small enough so that Bp(p) is a (spherical) conical neighborhood of p, then Sn n Bp(p)\{p } corresponds to the cone over S~ps. It then follows by induction on dim(B) that S s n A is a union of Amoa-walls for each apart- ment A C B. Consider an apartment A C B, and a pair of walls H1, He C A contained in S~. Lemma 3.7.2. -- If H~ is the image of H 2 under reflection in the wall H I (inside the apartment A), then H~ is contained in S B, Proof. -- To see this, consider an interior point p of a codimension 2 face a of H x n H e. The space Y.~ B decomposes as a metric join Z~ a o B~ where By is a 1-dimen- RIGIDITY OF Q UASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 141 Proposition 3.6.4. -- The space Z~B together with the collection of embeddings Z~o ~A : Y,~, S ~ Y~ B as above is a spherical building modelled on (Y'~0 S, W~). If ~ ~ B is an antipode orB, then we have a 1-1 correspondence between apartments (respectively half-apartments) in B containing {p,~ } and apartments (respectively half-apartments) in ~-,~ B; Y~ B /s thick provided B is thick. Proof. -- Any two points pq~, pq, ~ Z~ B lie in an apartment; namely choose qx, q2 close to p, then any apartment A containing ql, qe will contain p and p~, E Y,~ A. So SB1 holds. The space Z~ B satisfies SB2 since we are only using charts tA : S -+ B with tA(P0) = P and B itself satisfies SB2. The remaining assertions follow immediately from the definition of the spherical building structure on Y~ B. [] 3.7, Reducing to a thick building structure A reduction of the spherical building structure on B consists of a reflection subgroup W'C W and a subset ~r ~' which defines a spherical building structure modelled on (S, W'). The Amod-direction map 0 3 can then be factored as rc o 0~ where 0~ : B -~ W'\S =: a~o d is the A~oa-direction map for the building modelled on (S, W'), and : w'\s -- A% = W\S is the canonical surjection. Proposition 3."/.1. -- Let B be a spherical building modelled on the spherical Coxeter complex (S, W), with anisotropy polyhedron Amo d = W~S. Then there exists a reduction (W', ~') which is a thick building structure on B; W' is unique up to conjugacy in W and ~' is determined by W'. In particular, the thick reduction is unique up to equivalence, so the polyhedral structure is defined by the CAT(l) space itself. The proof will occupy the remainder of this paper. We set d =-dim(B), Rs ={p e B]Z~ B is isometric to a standard Sa-1}, and S~ = B\R s. If p E B and 0 ~ 0 is small enough so that Bp(p) is a (spherical) conical neighborhood of p, then Sn n Bp(p)\{p } corresponds to the cone over S~ps. It then follows by induction on dim(B) that S s n A is a union of Amoa-walls for each apart- ment A C B. Consider an apartment A C B, and a pair of walls H1, He C A contained in S~. Lemma 3.7.2. -- If H~ is the image of H 2 under reflection in the wall H I (inside the apartment A), then H~ is contained in S B, Proof. -- To see this, consider an interior point p of a codimension 2 face a of H x n H e. The space Y.~ B decomposes as a metric join Z~ a o B~ where By is a 1-dimen- 142 BRUCE KLEINER AND BERNHARD LEEB sional spherical building, and the walls Ha, H~, and H~ correspond to walls HI, H2, and H~ in B~; A corresponds to an apartment A in B~. The wall H x is just a pair of points in B~, and this pair of points is joined by at least three different semi-circles of length ~. These three semi-circles can be glued in pairs to form three different apartments in B~. Using the_ fact that an antipode_ of a point in SB, also lies in S~p, it is clear that the image of H 2 under reflection in H 1 is also in S~p. Hence the wall Y,~ H' 2 C Y,~ B is contained in three half-apartments, and proposition 3.6.4 then implies that H~ lies in three half-apartments. [] The reflections in the walls in A c~ S~ generate a group GA, and by [Hum, p. 24] the only reflections in G A are reflections in walls in A n S B; also, the closures of connected components of A\S~ are fundamental domains for the action of G A on A. Sublemma 3.7.3. -- Let U c B be a connected component 0fB\SB, and suppose U c~ A ~e 0 for some apartment A. Then U ~_ A. Proof. -- The set U c~ A is open and closed in U, so U c~ A = U. [] We claim that the isomorphism class of G A is independent of A. To show this, it suffices to show that the isometry type of a chamber Amo a is independent of A. For Ai i = 1, 2 let A~ be an apartment, and let Amo a be a chamber for GAi. If A 8 C B is an Ai apartment containing an interior point from each Amoa, then the sublemma gives Ai A~ C A 3. But then the A~o a are both chambers for GA3 , so they are isometric. Hence each pair (A, GA) is isomorphic to a fixed spherical Coxeter complex (S, W th) for some reflection subgroup W th ___ W. We denote the quotient map and model polyhedron by th 0 th : S ---> S/~ fth =: Amo d. We call the closure of components of B\Ss, tu Amoa-chambers. We can identify the A~oa-chambers with Amo th a in a consistent way by the following construction: Let A o __ B be an apartment and P0 ~ A0 n R B be a smooth point. We define the retraction p : B ~ A o by assigning to each point p in the open ball B~(P0 ) the unique point 9(P) e A o for which > ) the segments PoP and P0 P(P) have same length and direction PoP = Po P(P) atpo. The map p extends continuously to the discrete set B\B~(Po ) which maps to the antipode of Po in A 0. If A is an apartment passing through Po then A c~ Ao contains the A~o a- chamber spanned by P0 and PIA : A ~ A 0 is an isometry which preserves the tesselations by chambers. Composing p with the quotient map A o ~ A0/GAo we obtain a 1-Lipschitz map th (22) 0~ : B ---> Amid which restricts to an isometry on each chamber. Applying proposition 3.5.1 we see that B is a spherical building modelled on (S, W th) ; B is thick since we already verified in lemma 3.7.2 above that ifH C S~ is a wall, then it lies in at least three half-apartments. RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS t43 Corollary 3.7.4. -- For i = 1, 2 let B, be a thick spherical building modelled on (S,, W~) with atlas ~. If ~ : B 1 ---> B, is an isometry then we may identify the spherical Coxeter complexes by an isometry ~ : ($1, W1) -+ (S~, W2) so that ~ becomes an isomorphism of spherical buildings. 3.8. Combinatorial and geometric equivalences We recall (section 3.4) that for any building B, 9UB is the lattice of finite unions of faces of B. Proposition 3.8.1. -- Let B1, B 2 be spherical buildings of equal dimension. Then any lattice isomorphism JUB~ ~gf'B~ is induced by an isometry B~ -+ B~ of CAT(1)-spaces. This isometry is unique if the buildings B~ do not have a spherical de Rham factor. Proof. -- First recall that lattice isomorphisms preserve the partial ordering by inclusion since C 1 C C2 ~ C1 w C~ -= C~. We first assume that the buildings B~ have no de Rham factor and hence the JY'B~ come from simplicial complexes. In this case the lattice isomorphism ~Y'B1 --~ Jg'B~ carries k-dimensional faces of B1 to k-dimensional faces of B~. To see this, note that vertices of B~ are the minimal elements of the lattice 9t~B~ and k-simplices are characterized (induc- tively) as precisely those subcomplexes which contain k -? 1 vertices and are not contained in the union of lower dimensional simplices. Consider a codimension-2 face a of a chamber C in B~. For an interior point s e a, Z s B~ is isometric to the metric join ~ a o B7 where B~ is a 1-dimensional spherical building. The dihedral angle of C along a equals the length of a chamber in the 1-dimensional building B~. Sublemma 3. $. 2. -- The chamber length of a 1-dimensional spherical building is determined combinatoriatly as 2n/l where l # the combinatorial length of a minimal circuit. Proof. -- Combinatorial paths in a 1-dimensional spherical building determine geodesics. Closed geodesics in a CAT(1)-space have length at least 2~ since points at distance < ~ are joined by a unique geodesic segment. The closed paths of length 2~ are the apartments. [] Proof of Proposition 3.8.1 continued. -- As a consequence of the sublemma, the lattice isomorphism ~B1 ---> 9fiB2 induces a correspondence between chambers which preserves dihedral angles. Since the dihedral angles determine the isometry type of a spherical simplex [GrBe] [theorem 5.1.2], there is a unique map of CAT(1)-spaces B x -+ B~ which is isometric on chambers and induces the given combinatorial isomor- phlsm. Since the metric on each B~ is characterized as the largest metric for which the chamber inclusions are 1-Lipschitz maps, we conclude that our map B x -+ B~ is an isometry. In the general case, the buildings B, may have a spherical de Rham factor S~ and split as B~ = S~ o B~. The lattices 9t~B~ and 9t'B~ are isomorphic: to a subcomplex G" 144 BRUCE KLEINER AND BERNHARD LEEB of dr'B; corresponds the subcomplex S~o C; of vY'B~. The lattice isomorphism t ? ~B~ -~ ~Y'B x -+ ~B 2 ~ .YfB~ is induced by a unique isometry B 1 -+ B 2 by the discussion above. It follows that Dim B~ = Dim B' 2 and Dim S 1 ---- Dim $2. Any isometry $1 -+ Sz gives rise to an isometry B x -+ B 2 which induces the isomorphism ~/'B~ -+ ~B 2. [] 3.9. Geodesics, spheres, convex spherical subsets We call a subset of a CAT(1)-space convex if with every pair of points with distance less than ~ it contains the minimal geodesic segment joining them. The following generalizes corollary 3.5.2. Proposition 3.9.1. -- Let C C B a convex subset which is isometric to a convex subset of a unit sphere. Then G is contained in an apartment. Proof. -- We proceed by induction on the dimension of B. The claim is trivial if dim(B)= 0. We assume therefore that dim(B)> 0 and that our claim holds for buildings of smaller dimension than B. Let A be an apartment so that the number of open faces in A which have non-empty intersection with G is maximal. Suppose G ~ A. Let p e C n A and q e C\A be points with p~ r Y,~ A. Denote by V the union of all minimizing geodesics in A which connect p to its antipode i~ and intersect C --{p, i~ }; V is a convex subset of A and canonically isometric to the suspension of Y,~(C c~ A) = N~ C n ~ A. By the induction assumption, there is an apartment A' throughp such that Y,, C _ Z~ A'. A' can be chosen to contain/~. Then 121 n A _ V _ A' and p~ ~ Y~, A'. Hence the number of open sectors in A' inter- secting C is strictly bigger than the number of such sectors in A, a contradiction. There- fore C _ A. [] Corollary 3.9.2. -- Any minimizing geodesic in a spherical building B is contained in an apartment. Any isometrically embedded unit sphere K ~_ B is contained in an apartment. In particular dim(K) ~< rank(B) -- 1. 3.10. Convex sets and subbuildings A subbuilding is a subset B' __c_ B so that { ~ e d [ ~(S) =__ B' } forms an atlas for a spherical building structure; in particular B' is closed and convex. Lemma 3.10.1. -- Let s C B be a subset isometric to a standard sphere. Then the union B(s) of the apartments containing s is a subbuilding. There is a canonical reduction (W', ~') of the spherical building structure on B(s); its walls are precisely the W-walls of B(s) which contain s. When equipped with this building structure, B(s) decomposes as a join of s and another spherical building which we call Link(s). If p ~ s then log s maps Link(s) isometrically to the join complement of Y,p s in Z~ B(s). Furthermore, if p e s lies in a W-face a of maximal possible dimension, then there is a bijective correspondence between W-chambers containing a, W'-chambers of B(s), chambers of Link(s), and W,,-chambers in Z,~ B. RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS t45 Proof. -- Let 4 and ~ be interior points of faces in s with maximal dimension. Then B(s) is the union of all geodesic segments of length = from 4 to ~. Proposition 3.6.4 implies that every pair of points in B(s) is contained in an apartment A C B(s). Pick ~o e d with s_~ ~0(S), and set d' -~ {~ e d I~Is ~ : ~olso}- Let W' _~ W be the subgroup generated by reflections fixing s o pointwise. According to lemma 3.1.1, the coordinate changes for the charts in d' are restrictions of elements of W'. Therefore ~' is an atlas for a spherical building structure on B(s) modelled on (S, W'). Since s o ___ S is a join factor of the spherical Coxeter complex (S, W'), B(s) decomposes as a join of spherical buildings B(s) = s o Link(s) by section 3.3. Any two points in Link(s) lie in an apartment s _~ A C B(s), so log~ maps Link(s) isometrically to the join complement of 5] s in X~ B(s). The remaining statements follow. [] The building B(s) splits as a spherical join of the singular sphere s and a spherical building which we denote by Link(s): B(s) = s o Link(s). Lemma 3.10.2. -- If 4 e B and ~ lies in the apartment A ~_ B, then there is a ~ e A with ~ ~ d(4, ~) = d(~, ~) + dOq , ~). If d(4, ~) >t ~/2 then ~ has an antipode in every top- dimensional hemisphere H C A. Proof. -- When Dim B = 0 the lemma is immediate. Ifd(~, n) < 7: then by induction e X, B has an antipode in Zn A. Therefore we may extend 4~ to a geodesic segment 4~ with ~ C A of length ~. The second statement follows by letting ~ be the pole of the hemisphere. [] Proposition 3.10.3. -- Let C be a convex subset in the spherical building B. If C contains an apartment then C is a subbuilding of fuU rank. Proof. -- By tile lemma, any point ~ e C has all antipode ~ in C. By lemma 3.6.1, the union C~, ~ of all minimizing geodesics from ~ to ~ which intersect C --{ 4, ~ } is a neighborhood of 4 in C. In particular, for sufficiently small e > 0, C n B,(4) is a cone over Y.~ C. Since ~ can be chosen to lie in an apartment A 0 _ C by our assumption, and since the apartment Y,~A 0 in Z~ C corresponds to an apartment in Cr we see that C is a union of apartments. It remains to check that any two points 4, ~ e C lie in an apartment contained in C. Choose an apartment A with ~ e A _ C. For ~4 e Y,~ C there exists an antipodal direction in ~ A and we can extend 4~ into A to a geodesic 4~4 of length ~. To the apartment ZpA in ~ C corresponds an apartment A'_ C~,~ containing 4~. [] 3.11. Build~ng morphisms We call a map ? : B ---> B' between buildings of equal dimension a building morphism if it is isometric on chambers. Later, when looking at EucIidean buildings, we will encounter natural examples of building morphisms, namely the canonical maps from the Tits boundary to the spaces of directions. 19 146 BRUCE KLEINER AND BERNHARD LEEB A building morphism r has Lipschitz constant 1:q0 maps sufficiently short segments emanating from a point p isometrically to geodesic segments. Therefore it induces well- defined maps (23) Z~ ~ : Z~ B -+ Z~l B'. Since the chambers in B containing p correspond to the chambers in Z~ B (with respect to its natural induced building structure, cf. Proposition 3.6.4), and similarly for B', the maps (23) are building morphisms, as well. We call the morphism q0 spreading if there is an apartment A o __ B so that VlAo is an isometry. Lemma 3.11.1. -- Let q0 : B -+ B' be a spreading building morphism. Then, if ~l, 4z e B are points with q~(41) = q~(~2) =: 4', the images of Z~l ~ and ~2 q~ in ~, B" coincide. Proof. -- If q~ is spreading then each point ~' e ~(B) has an antipode ~' e ~(B). Any points 4 e q~-l(4 ') and ~ ~q~-l(g,) are antipodes and minimizing geodesics connecting ~ and ~ are mapped isometrically to geodesics connecting 4' and ~', i.e. ~{B,~,~, : B(4, ~) -+ B'(~', ~') is the spherical suspension of the morphism Z~ q~. There are canonical isometries persp~.~ : Z~ B -+ Y.2 B and persp~, 2, : Z~, B' -+ Z~, B', cf. 3.6.1, and we have: (24) Z~ ~ o persp~,'~ = persp~,.~, o Y~ ~. The assertion follows. [] Lemma 3.11.2. -- Let ~ :B ~B' be a spreading building morphism. Suppose 41eB, t t ~2 ~ B' and set 4~ := r Then there is an apartment A ~ B containing ~1 such that cOlA is an isometry and the apartment A' := ~A _c B' contains ~2. Proof. -- Let us first assume that ~'2 ~ A~ ----- q0A 2 where A, is an apartment in B such that q0{~, is an isometry. Then there is a geodesic segment ~ 42 41 of length n such that 42'^'41C a 2' (lemma 3.10.2). Let gl cA, be the lift of ~'~. By proposition 3.6.4, the subbuilding B(~, ~) contains an apartment A with Z~ A = Z~ A,. The map r is an isometry, because it is an isometry near ~. By construction, 4~ e q~A. The above argument implies that, since q0 is spreading by assumption, each point ~x e B lies in an apartment A~ so that q0]** is an isometry. Therefore the assumption in the beginning of the proof is always satisfied and the proof is complete. [] Corollary 3.11.3. -- Let ~ be as in lemma 3.11.2. Then: 1. r is a subbuilding in B'. 2. The induced morphisms Y,~ ~ are spreading. 3. For all ~ e B, ~ e ~(B), there exists ~ e ~-1 ~'2 suck that (25) d~(4~, 42) = dB,(q~41, 4~). 4. If 42 satisfies (25) then there exists an apartment A ~_ B containing 4~ , 4~ such that r ]~ is an isometry. RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 147 Proof. -- The first three assertions follow immediately from the lemma. We prove the fourth assertion: t t At By 1. we find a geodesic segment 41 4.2 41 of length rc contained in ?(B). By 3. there exists a lift ~1 of ~ such that d~(42, ~1)= dB,(4'2, ~'~). Applying the previous lemma to the morphism Z~I ?, which is spreading by 2., we find an apartment A _~ B(41, ~1) containing the geodesic segment 41 ~2 ~1 and so that X~, ?]~ a, and therefore also ?tA, is an isometry. [] Proposition 3.11.4. -- Let B and B' be spherical buildings modelled on Amoa, and let ? : B -+ B' be a surjective morphism of spherical buildings so that 0 B = O n, o ?. Suppose v is a face orB and 6' is a face orB' contained in ?(B) so that ?v ~_ 6'. Then there exists a face 6 of B with "~ ~_ 6 and 76 = 6'. Proof. -- Let 4 be an interior point of v and let 61 be a face of B with ?61 = 6'. Then 61 contains (in its boundary) a point 41 with ?41 ---- ?4, and by lemma 3.11.1 there exists a face 6 containing 4 (and therefore v) with ?6 ---- q~al = 6'. [] Corollary 3.11.5. -- Let B, B' and ? be as in proposition 3.11.4. If h' C B' is a half- apartment with wall m', and m C B lifts m', then there is a half-apartment h C B containing m which lifts h'. Proof. -- Let v'C h' be a chamber with a panel 6'C m', and let 6 C m be the lift of 6' in m. Applying proposition 3.11.4 we get a chamber v C B so that the half-apart- ment h spanned by v u m lifts h'. [] 3.12. Root groups and Moufang spherical buildings A good reference for the material in this section is [Ron]. Definition 3.12.1 ([Ron, p. 66]). -- Let (B, Amod) be a spherical building, and let a C B be a root. The root group U a of a is defined as the subgroup of Aut(B, Amoa) consisting of all automorphisms g which fix every chamber C C B with the property that C c~ a contains a panel ~ r Oa. We let G B C Aut(B, Amod) be the subgroup generated by all the root groups of B. Proposition 3.12.2 (Properties of root groups). -- Let B be a thick spherical building. 1. If U a acts transitively on the apartments containing a for every root a contained in some apart- ment A0, then the group generated by these root groups acts transitively on pairs(C, A) where C is a chamber in an apartment A ~_ B. 2. Suppose (B, Amod) is irreducible and has dimension at least 1. Then the only root group element g e Ua which fixes an apartment containing a is the identity. 148 BRUCE KLEINER AND BERNHARD LEEB Lemma 3.19..3. -- Let A and A' be apartments in the spherical building B. Then there exist apartments A 0 = A, Ax, ..., A k = A' so that A~_ ~ n A~. is a half-apartment containing A c~A'for all i. Proof. -- Suppose that A and A' are apartments which do not satisfy the conclusion of the lemma and so that the complex A n A' has the maximal possible number of faces. We derive a contradiction by constructing an apartment A" whose intersection with A respectively A' strictly contains A n A'. If A n A' is empty, we choose A" to be any apartment which has non-empty intersection with both A and A'. If A n A' is contained in a singular sphere s of dimension dim(A c~A') < dim(B) we pick a chambers 6C A and u'C A' with dim(u n s)= dim(u' n s)= dims. The subbuilding B(s) contains an apartment A" with s t_) 6 u ~' C A" and A" has the desired property. It remains to consider the case that A n A' contains chambers and is strictly contained in a half-apartment. Then there is a half-apartment h C A containing A n A' and so that Ok n A n A' contains a panel re. Let 6' C A' be a chamber with 6' n A n A' = r~. The convex hull A" of h u u' is an apartment with the desired property. [] Proof of Proposition. -- 1. Let G x be the group generated by the root groups U~ where a runs through all roots contained in an apartment A C B. If g ~ U~ then G A = GaA because U,x = gU, g -1 for all roots a C A. By lemma 3.12.3, given any apartment A' there is a sequence A0, ... , A k = A' such that A~._ a n A~. is a root. Hence GA0 = GA1 ..... G x, and it follows that G~ ----- G A, for all apartments A'. Let 6x and 62 be chambers in B which share a panel rc = ux n u2. Since B is thick, there is a third chamber 6 with 6 n 6~ = n. Pick apartments A~. containing u u 6~. Applying lemma 3.12.3 again, we see that there is a g ~ G~ so that g(Ax) = As, and g fixes u s. Hence gux = us and we conclude by induction that G B acts transitively on chambers. Let A1, A z be apartments and u1, u2 be chambers such that u~ ___ A~.. By the above argument, there exists g ~ G B with gux = u~. By lemma 3.12.3 there is a g' ~ G B with g'(gA1) ---- A~ and g' 6s = 6s. Hence GB acts transitively on pairs C C A as claimed. 2. Since B is irreducible, there is a chamber u contained in the interior of a (see lemma 3.3.2). Since the convex set B' = Fix(g) contains the apartment A it is a sub- building by proposition 3.10.3. Moreover, B' contains an open neighborhood of u by the definition of U a. Note that if ~ and n' are opposite panels in B', then B' contains every chamber containing n if and only if it contains every chamber containing n' (lemma 3.6.1). Since for each panel ~ there is a panel ~x C 0u in the same projectivity class (see definition 3.6.2 and lemma 3.6.3) we see that B' contains every chamber in B with a panel in B'. When Dim(B) = Dim(B') = 1 this implies that B' is open in B, forcing B' = B; in general we show by induction that for all p e B' we have Y~ B' = Y~ B, which implies that B'C B is open and consequently B' = B. [] RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 149 Definition 3.12.4. -- A spherical building (B, A~o~) /s Moufang if for eack root a C B the root group Uo acts transitively on the apartments containing the root a. When B is irreducible and has rank at least 2, then by 2. above, U, acts simply transitively on apartments containing a. The spherical building associated with a reductive algebraic group ([Ti 1, chapter 5]) is Moufang. In particular, irreducible spherical buildings of dimension at least 2 are Moufang. 4. EUCLIDEAN BUILDINGS There are many different ways to axiomatize Euclidean buildings. For us, the key geometric ingredient is an assignment of Amoa-directions to geodesics segments in a Hadamard space. Just as with symmetric spaces, ~ocdirections capture the anisotropy of the space, and they behave nicely with respect to geometric limiting operations such as ultralimits, Tits boundaries, and spaces of directions. 4.1. Def~nltlon of Euclidean buildings 4.1.1. Euclidean Coxeter complexes Let E be a finite-dimensional Euclidean space. Its Tits boundary is a round sphere and there is a canonical homomorphism (26) p : Isom(E) -+ Isom(0~l~, E) which assigns to each affine isometry its rotational part. We call a subgroup War C Isom(E) an affine Weyl group if it is generated by reflections and if the reflection group W := ?(War) C Isom(0~l ~ E) is finite. The pair (E, War) is said to be a Euclidean Coxeter complex and (27) 0~,~(E, War) := (0~,~ E, W) is called its spherical Coxeter complex at infinity. Its anisotropy polyhedron is the spherical dolyhedron := E)/W. An oriented geodesic segment ~ in a E determines a point in 0~l ~ E and we call its projection to Amo d the Amoa-direction of ~. A wall is a hyperplane which occurs as the fixed point set of a reflection in War and singular subspaces are defined as intersections of walls. A half-space bounded by a wall is called singular or a half-apartment. An intersection of half-apartments is a Weyl- polyhedron. Weyl cones with tip at a point p are complete cones with tip at p for which the boundary at infinity is a single face in 0~l ~ E. Fix a point p ~ E. By W(p), we denote the subgroup of War which is generated by reflections in the walls passing through p; W(p) embeds via 0 as a subgroup of W. 150 BRUCE KLEINER AND BERNHARD LEEB A Weyl sector with tip at p is a Weyl polyhedron for the Euclidean Coxeter complex (E, W(p)); note that a Weyl sector need not be a Weyl cone, and a Weyl cone need not be a Weyl sector. A subsector of a sector e is a sector e' C ~ with 0~t, ~' = 0~t~ ~; lies in a finite tubular neighborhood of e'. A Weyl chamber is a Weyl polyhedron for which the boundary at infinity is a Amo d chamber; Weyl chambers are necessarily Weyl cones. The Coxeter group W(p) acts on X~ E, so we have a Coxeter complex E,(E, W~,,) := (Z, E, W(p)) with anisotropy map by 0, : Z~ E ~ Y., E/W(p) =: A~od(P). The faces in (Y,~ E, W(p)) correspond to the Weyl sectors of E with tip at p. We call the Coxeter complex (E, W,xf) irreducible if and only if its anisotropy polyhedron, or equivalently, its spherical Coxeter complex at infinity is irreducible. In this case, the action of W on the translation subgroup T <1 War forces T to be trivial, a lattice, or a dense subgroup. In the latter case we say that Wm is topologically transitive. 4.1.2. The Euclidean building axioms Let (E, Waft) be a Euclidean Coxeter complex. A Euclidean building modelled on (E, War ) is a Hadamard space X endowed with the structure described in the following axioms. EB1. Directions. -- To each nontrivial oriented segment ~ C X is assigned a Amo d- direction 0(x-y) e Amo a. The difference in Amod-direetions of two segments emanating from the same point is less than their comparison angle, i.e. (28) d(0(~), 0(x-z)) ~< ~_,(y, z). Recall that given 81, 82 e Amod, D(81, 8~) is the finite set of possible distances between points in the Weyl group orbits 0~-T]t,~.(81) and 0~-T]t,~.(8~). EB2. Angle rigidity. -- The angle between two geodesic segments ~ and ~ lies in the finite set D(O(~), 0(~)). We assume that there is given a collection d of isometric embeddings ~ : E -+ X which preserve Amod-directions and which is closed under precomposition with isometrics in War. These isometric embeddings are called charts, their images apartments, and A is called the atlas of the Euclidean building. EB3. Plenty of apartments. -- Each segment, ray and geodesic is contained in an apartment. The Euclidean coordinate chart tA for an apartment A is well-defined up to pre- composition with an isometry e e O-I(W). Two charts ~A1, ~A2 for apartments A1, A 2 are said to be compatible if t-all ~ t.~ is the restriction of an isometry in W~u. This holds automatically when Waf f = p-l(W). RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 151 EB4. Compatibility of apartments. -- The Euclidean coordinate charts for the apartments in X are compatible. It wilt be a consequence of Corollary 4.6.2 below that the atlas sJ is maximal among collections of charts satisfying axioms EB3 and EB4. We define walls, singular flats, half-apartments, Weyl cones, Weyl sectors, and Weyl polyhedra in the Euclidean building to be the images of the corresponding objects in the Euclidean Coxeter complex under charts. The set of Weyl cones with tip at a point x will be denoted by ~. The rank of the Euclidean building X is defined to be the dimension of its apartments. The building X is thick if each wall bounds at least 3 half-apartments with disjoint interiors. We call X a Euclidean ruin if its underlying set or the atlas ~' is empty. 4L 1.3. Some immediate consequences of the axioms Axiom EB1 implies the following compatibility properties for the Amoa-directions of geodesic segments. Lemma 4.1.1. -- Let x, y, z be points in X. 1. If y lies on -~, then 0(~) = 0(~) = 0()-~). 2. If ~y, ~z e Z, X coincide, then 0(~) = 0(~). 3. Asymptotic geodeaic rays in X have the same Amoa-direction. We call a segment, ray or geodesic in X regular if its Amoa-direction is an interior point of Amo a. Lemma 4.1.2. -- 1. If p e X and x~ e X --p, then the px---] initially span a flat triangle /f/~(xx, x2) > 0, and they initially coincide if/-~(xl, x2) = O. 2. If p~ ~ X and ~ e OT~ X, then the rays p~ ~ are asymptotic to the edges of a flat sector. Proof. -- 1. After extending the segments p-k~ to rays if necessary, we may assume without loss of generality that x, ~ Oxi~X. If z epx~, then 0(~)= 0(xi) so Z-,(xl, x~) a D(0(xa) , 0(x~)) which is a finite set. But /_,(xl, xa) -+/-~(xl, x,.) mono- tonically as z ~ p, which implies that/__,(Xl, x2) =/-,(xx, Xz),/,(P, xz) = ~ -- /~(xl, x2) when z is sufficiently close to p. Therefore A(p, z, x~) is a flat triangle (with a vertex at oo) when z is sufficiently close to p. 2. follows from similar reasoning and the property (6) of the Tits distance. [] 4.2. Associated spherical building structures 4.2.t. The Tits boundary The Tits boundary 0~l ~ X is a CAT(1)-space, see 2.3.2. Lemma 4.1.1 implies that there is a well-defined Amoa-direction map (29) 00TitsX -* 0Tlta g ---> Alnod which is 1-Lipschitz by (28). 152 BRUCE KLEINER AND BERNHARD LEEB Proposition 4.2. t. -- The space O~lt8 X carries a spherical building structure modelled on the spherical Coxeter complex (0~it~ E, W) with A.,oa-direction map (29). Proof. -- We verify that the assumptions of proposition 3.5.1 are satisfied. Axiom EB2 implies that (29) satisfies the discreteness condition (17). IrA is a Euclidean apartment in X then 0Tit. A is a standard sphere in 0~lt, X. Clearly, any point ~ ~ 0~lts X lies in a standard sphere. It remains to check that any two points ~1 and ~2 in 0~l ~ X with Tits distance rc are ideal endpoints of a geodesic in X. To see this, pick p ~ X and note that the angle /~(~1, ~) increases monotonically as z moves along the ray P~I towards ~1. But by EB2 /~(~1, ~2) assumes only finitely many values, so when z is sufficiently far out we have /~z(~l, ~2) = /Tita(~l, ~2) = 717, and the rays z~, fit together to form a geodesic with ideal endpoints ~1 and ~2. [] 4.2.2, The space of directions The space of directions E~ X is a CAT(1)-space (see section 2.1.3). Lemma 4.1.1 implies that there is a well-defined 1-Lipschitz map from the space of germs of segments in a point x eX: (30) 0zx x : Y,~ X ~ Amo d. In this section we check that this map induces a spherical building structure on ~ X. By axiom EB2, 0 = 0zx x satisfies the discreteness condition (17). Lemma 4.2.2. -- The space X~ X is complete, so E* X = X~ X. Proof. -- Let (x~) be a sequence in X -- { x } such that (x-~k) is Cauchy in Y~*, X. Then 0(x-kk) is Cauchy in Amo a and we denote its limit by ~. If A k C X is an apartment containing x--x k then x-k~ e E~ A k C E~ X and E, A k contains a spherical polyhedron % such that x~ ~ % and 01ok; %-+ A~o a is an isometry. There is a unique ~ ~ % with 0(~k) = ~ and we have d(~, x~) = damoa(~ , 0(x-~k) ) -+ 0. Hence (~) is Cauchy with 0(~) = 8 and lim x~ = lim ~ in Y~ X. The discreteness condition (17) implies that (~) is eventually constant and therefore (x-~) has a limit in E~ X. [] We now apply proposition 3.5.1 to verify that E, X carries a natural structure as a spherical building modelled on (0Tl ~ E, W). The only condition which remains to be checked is that antipodal points x~ and x~ z in Y~ X lie in a subset isometric to S = 0TIt~E. But /~(x~, x2)= 7: implies that x~x~ = xx---l~ w xxa and if A C X is an apartment containing x~ x2 then E~ A C E, X is a spherical apartment containing x~ a and ~z. Lemma 4.9~.3. -- All standard spheres in Y~, X are of the form Y~,, A where A is an apartment in X passing through x. Proof. -- By corollary 3.9.2, standard spheres are A~oa-apartments, so we can find antipodal regular points ~1, ~z e ~. Then there is a segment Xl xz through x with RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 153 x~i =- ~i. If A ~ X is an apartment containing xl x2 then E, A c~ ~ ~ { ~1, ~. } and the spherical apartments e and E, A coincide because they share a pair of regular antipodes (lemma 3.6.1). [] There are two natural reductions of the Weyl group which we shall consider. First, according to section 3.7 there is a thick spherical building structure with atlas ~th(x) and anisotropy map (31) : z. x -+ This structure is unique up to equivalence. The second reduction is analogous to the structure constructed in proposition 3.6.4. We postpone discussion of this structure until 4.4.1 because we do not have an analog of lemma 3.1.1 in the case of nondiscrete Euclidean Coxeter complexes. 4.3. Product (-decomposition)s Let X~, i = 1,..., n, be Euclidean buildings modelled on Coxeter complexes (E,, W~r) with atlases ~r and anisotropy polyhedra A~o a. Then W~:= W~ x ... x W2e acts canonically as a reflection group on E := E~ � ... � E,. We call the Coxeter complex (E, Wan ) the product of the Coxeter complexes (E~, W~) and write (32) (E, = (Ex, WI ,) x ... x (E,, There are corresponding join decompositions (33) (6,~ E, W) = (6i~ El, Wx) o ... o (6,~ E,, W,) of the spherical Coxeter complex at infinity and (34) Amo a = Amo a o ... o A~o a of the anisotropy polyhedron. The Hadamard space (35) X = Xl x ... x X, carries a natural Euclidean building structure modelled on (E, War ). The charts for its atlas .~ are the products t = q x ... � t, of charts q e d,. We call X equipped with this building structure the Euclidean building product of the buildings X~.. Proposition 4.8.1. -- Let X be a Euclidean building modelled on the Coxeter complex (E, W~) with atlas d and assume that there is a join decomposition (34) of its anisotropy polyhedron. Then 1. There is a decomposition (32) of (E, W~) as a product of Euclid~an Coxeter complexes so that a segment -~ C E is parallel to the factor E~ if and only if its Amod-direction 0(~) lies in &,oa. 2. There is a decomposition (35) of X as a product of Euclidean buildings so that a segment xy- C E is parallel to the factor E~ if and only if its Amoa-direction 0(-~) lies in Amo a' . 20 154 BRUCE KLEINER AND BERNHARD LEEB Proof. -- 1. Proposition 3.3.1 implies that tile spherical Coxeter complex at infinity decomposes as a join (36) E, W) = (Sl, Wl) o ... o of spherical Coxeter complexes. By proposition 2.3.7, this decomposition is induced by a metric product decomposition E = E i � ... � E~ so that 0Ti mE, is cano- nically identified with Si and, hence, a segment ~ C E is parallel to the factor E, if and only if 0(~)eA~o a. (36) implies that War f decomposes as the product Waff= W 1 � ... � W2e r of reflection groups W*~u acting on E,, thus establishing the desired decomposition (32). 2. Arguing as in the proof of the first part, we obtain a metric decomposition (35) as a product of Hadamard spaces so that ~ C X is parallel to the factor X~ if and only if 0(~) e A~o a. Furthermore, the 0Ti ~ X~ carry spherical building structures modelled on (0Tit~E~, W~) so that the spherical building 0Ti ~ X decomposes as the spherical building join of the 0Tl ~ Xi. Each chart ~ : E -+ X, ~ e d, decomposes as a product of A~oa-direction preserving isometric embeddings ~ : Ei ~ X~.. The collection d~ of all ~i arising in this way forms an atlas for a Euclidean building structure on X~. and (35) becomes a decomposition as a product of Euclidean buildings. [] We call a Euclidean building irreducible if its anisotropy polyhedron is irreducible, compare section 3.3. According to the previous proposition, the unique minimal join decomposition of the anisotropy polyhedron Amo a into irreducible factors corresponds to unique minimal product decompositions of the Euclidean Coxeter complex (E, W~) and the Euclidean building X into irreducible factors. We call these decompositions the de Rham decompositions and the maximal Euclidean factors with trivial affine Weyl group the Euclidean de Rham factors. 4.4. The local behavior of Weyl-cones In this section we study the set :r of Weyl cones with tip at p. The main result (corollary 4.4.3) is that in a sufficiently small neighborhood of p, a finite union of these cones is isometric to the metric cone over the corresponding finite union of A~o a faces in Z~ X. This proposition plays an important role in section 6. Let Wi and W~ be Weyl cones in X with tip at p. The Weyl cone W i determines a face Z~ W~ in the spherical building (Z~ X, A~oa). Sublemma 4.4.1. -- Suppose that Z~ Wi = E~ W, in Z~ X. Then Win W, is a neighborhood of p in W i and W2. Proof. -- According to lemma 4.1.2 each point in the face E~ W i = E~ Wz is the direction of a segment in W1 n W 2 which starts at p. We can pick finitely many points in Y,~ W~ = Z~ W o whose convex hull is the whole face. The convex hull of the corresponding segments is contained in the convex set W1 n W, and is a neighborhood ofp in W1 and W 2. [] RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 155 Locally the intersection of Weyl cones with tip at a point p is given by their infinitesimal intersection in the space of directions Z~ X: cone W 9 ~lt/" with Lemma 4.4.9~. -- If Wx, Wz 9 then there is a Weyl Z~ W -= Z~ W 1 n X~ Wz. For every such W there is an ~ > 0 so that W 1 n W, n B,(p) = W c~ B,(p). Hence the intersection of Weyl cones with tip at the same point is locally a Weyl cone. Proof. -- By lemma 3.4.2 the intersection Y.= Wx c~ Y,~ W~ is a A~oa-face and hence there is a W 9 W'~ such that ~ W = X~ W~ c~ E~ Wz. By the previous sublemma, there are W~ e W'~ with W~ c Wi and a positive 9 so that W 1 /'~ gg(p) = W~ ~ Bl:(p ) = W f'~ gg(p) for any such W. If x is a point in W 1 n Wz different from p then p~ 9 X~ W, so p-x C W'a t3 W'~. Therefore t t W x ~ W~ n B~(p) = W~ c~ W~ ~ B,(p) = W r~ B,(p). [] Corollary 4.4.3. -- If W1,...,Wk 9 then there is an ~>0 such that (U, W,) c~ B~(r maps isomarically to (U~ c~ w~) n B(~) c C~ X via log~. Proof. -- Let 5 denote the finite subcomplex of Y,~ X determined by Ui Y~f W,. Pick ,~, "2 9 5. By lemma 4.2.3 these lie in an apartment X~ A,to~ c y, X for some apartment Ao~ ~ C X passing through p. If *x is a face of ~ W~ and ~2 is a face of Y,~ Wj, then by the sublemma above we may assume without loss of generality that (W~ ~ u W~') n B~(~) _ Ao~o, where W~ t (resp. W~) is the subcone of W~ (resp. W,) with Z~ W~ ----- ex (resp. ~= W~ ----- a2). Since there are only finitely many such pairs el, a9 9 5, for sufficiently small r > 0, every pair of segments px--~,px--~ __ U~ W~ bounds a fiat triangle provided ]px,] < ~. [] 4.4.1. Another building structure on Y.~ X, and the local behavior of Weyl sectors Let ~ C E X be a &~oa-apartment. By lemma 4.2.3 there is an apartment A C X with Y~ X = ,t, and by corollary 4.4.3 any two such apartments coincide near p. Hence the walls in A which pass through p define a reflection group W~ C Isom(~). Lemma 4.4.4. -- The reflection group W~ contains the reflection group W~ coming from the thick spherical building structure on l~ X. Proof. -- Let m C 0r be a wall for the A~oa(p) structure. There are apartments A, C X through p, i -= 1, 2, 3, so that X~ A t = ,t and the ~ Ph intersect in half-apart- ments with boundary wall m. By corollary 4.4.3 the pairwise intersections of the A~ are half-spaces near p. Choose charts Lxt , tat , tA, 9 d and let ~ 9 War be the unique 156 BRUCE KLEINER AND BERNHARD LEEB isometry inducing ~/o tAi. Then qhs o q~2s o %~ is a reflection at a wall w passing through x = t~xl(p) and satisfying ~ ~A1 w -= m. [] Fixing one apartment ~ C Z~ X, we take a chart t : S -+ e from the atlas s~t~(p), and enlarge ~a(p) by precomposing each chart ~' e~Cth(p) with elements of t, ~(W~) C Isom(S). Clearly this defines an atlas ~r for a spherical building structure modelled on A~oa(p) aof cr Let A, A 1 C X be apartments so that ~ A = ~, Z~ Az = ~1, and e ca cr 1 contains a chamber C C r162 If ~x, ~a, : E -+ X are charts from the atlas ~r then since A ca A 1 is a cone near p by lemma 4.4.3, it follows that Z~(~al o t2 ~) : g~ A = ~ --> al = Z~ A~ carries W~ faces in a to W~ faces in e~, while at the same time it carries Amoa(p) faces of ~ to Amoa(p) faces of ~1. So every A~od(p) face ~ C Z~ X is a W~, face for every apartment r162 containing a. Since the W~,'s are all isomorphic, this clearly implies that Z~ W is a Amoa(p) face for every Weyl sector with tip at p. So we have shown: Proposition 4.4.5. -- There is a spherical building structure (Y,~ X, d(p)) modelled on (S, Amod(p) ) so that Amoa(p)-faces in F,~ X correspond b~fectively to the spaces of directions of Weyl sectors with tip at p. In particular, if A C X is any apartment passing through p, then there is a 1-1 correspondence between walls m C A passing through p and Amoa(p)-walls in the apart- ment F,~ A, given by m ~-. Z~ m. When X is a thick building, then ~f(p) coincides with ~h(p) for every p E X. Corollary 4.4.6. -- Corollary 4.4. S holds when the W, are Weyl sectors with tip at p. and A 2 be apartments in X then A 1 ca A t is either empty or a Weyl polyhedron. ]n particular, If A1 ca A S contains a complete regular geodesic then A 1 = A S. A1 Proof. -- Each Weyl sector with tip at p is a finite union of Weyl cones with tip at p. Hence a finite union of Weyl sectors with tip at p is a finite union of Weyl cones with tip at p, and the first statement follows. If A1, A S C X are apartments and p e A 1 ca As, then Z~ Q1 ca z~ A 2 is a convex Amoa(p) subcomplex ofY.~ A~. Hence there are Amod(p) half apartments hl, ..., hk C Y,~ A 1 so that ['1, h, = Y.~ AI ca X~ A S. By proposition 4.4.5, for each i there is a half-apartment H, C A with Z~ H~ = h~. Therefore A 1 ca A s ca B~(s) = (N Hi) n B~(r and so A x ca A, is a Weyl polyhedron near p. Consequently A 1 t~ A, is a Weyl polyhedron. [] 4.5. Discrete Euclidean build;rigs We call the Euclidean building X discrete if the affine Weyl group W~ is discrete or, equivalently, if the collection of walls in the Euclidean Coxeter complex E is locally finite. If p is a point in E then % denotes the intersection of all closed half-apartments containing p, i.e. the smallest Weyl polyhedron containing p. By corollary 4.4.6, each affine coordinate chart ~A : E -+ X maps % to the minimal Weyl polyhedron in X which RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 157 contains ~A(P)- Hence for any point x a X there is a minimal Weyl polyhedron % containing it. We say that x spans %. % is the intersection of M1 half-apartments containing x and, if X is thick, the intersection of all such apartments. The lattice of Weyl poly- hedra % with x e % is isomorphic to the polyhedral complex Jf'E, X, Proposition 4.5.1. -- In a discrete Euclidean building X each point x has a neighbor- hood B,(x) which is canonically isometric to the truncated Euclidean cone of height ~ over ~ X. Proof. -- Let tA : E -+ X be a chart with x = t A(p) and choose a > 0 so that any wall intersecting B,(p) contains p. Then for any point y e B.(p), the polyhedron % contains x and any apartment intersecting B,(p) passes through x. Hence any two segments ~ and ~ of length < , lie in a common apartment and it follows that B,(p) is isometric to a truncated cone. [] Assume now that W~ff is discrete and cocompact. Then the walls partition E into polysimplices which are fundamental domains for the action of W,f e. This induces on X a structure as a polysimplicial complex. The polysimplices are spanned by their interior points. If X is moreover irreducible, then this complex is a simplicial complex. 4.6. Flats and apartments Proposition 4.6.1. -- Any flat F in X is contained in an apartment. In particular, the dimension of a flat is less or equal to the rank of X. Proof. -- Among the faces in 0Ti ~ X which intersect the sphere 0Tit~ F we pick a face e of maximal dimension. Then ~ n 0~lt~ F is open in 0Tit~ F, Let c be a geodesic in F with c(oo) ~ Int(a) and let A be an apartment containing c. Then 0Tlt~ A contains and c(-- oo) and convexity implies 0Tit, F __q 0~lt, A. Since F n A 4~ 0, it follows that F is contained in the apartment A. [] As a consequence, we obtain the following geometric characterization of apartments in Euclidean buildings: Corollary 4.6.2. -- The r-flats in X are precisely the apartments. The next lemma says that a regular ray which stays at finite Hausdorff distance from an apartment approaches this apartment at a certain minimal rate given by the extent of its regularity. Lemma 4.6.3. -- Suppose ~ ~ O, lt, X is regular and that the ray p~ remains at bounded distance from an apartment F. Then every point x ~ p~ with d(p, F) d(x, p) lies in F. 158 BRUCE KLEINER AND BERNHARD LEEB Proof. -- Lety be a point on the ray 7~A(p) ~, and let z ~fi~ be the point where the segment ~ enters A (we may have z =y). By lemma 4.1.2 /,(p, A) > 0, and by lemma 3.4.1 we have/,(p, A) 1> da~od(0(fi~), 0 Amoa). The comparison triangle A(a, b, c) in the Euclidean plane for the triangle A(p, teA(p) , z) satisfies /b(a, c)>1 ~/2 and /,(a, b) t> dA~od(0(fi-~), 0 Amod). Hence d(p, A)/> d(p, z) sin(da~od(0(fi~), 0 A~od) ). Since 0(fi-5) = 0(~) ---> 0(p~) as y ~p~ tends to ~, the claim follows. [] Corollary 4.6.4. -- Each complete regular geodesic which lies in a tubular neighborhood of an apartment A must be contained in A. IrA 1 and A 2 are apartments in X and A 2 lies in a tubular neighborhood of A1, then A 1 : A s. Another implication of the previous lemma is the following analogue oflemma 4.4.2 at infinity. Lemma 4.6.5. -- If C1, C2 C X are Weyl chambers with OTlt~ C a : Ozit~ C2, then there is a chamber C ~ C 1 n C~. Proof. -- It is enough to consider the case that the building X is irreducible. The claim is trivial if the affine Weyl group is finite and we can hence assume that Wa~ is cocompact. If p is a regular geodesic ray in C1 then, by the previous lemma, it enters C, in some pointp and C1 r~ C, contains the metric cone K centered atp with ideal boundary O~l ~ K = 0~its CA. Since W~ is cocompact, K clearly contains a Weyl chamber. [] Proposition 4.6.6. -- There is a bijective correspondence between apartments in X and OTlt~ X given by A ~_ X ~ 0~it. A c_ 0Tlt~ X. Proof. -- We have to show that every apartment K in OTl ~ X is the boundary of a unique apartment in X. Since K contains a pair of regular antipodal points, there is a regular geodesic c whose ideal endpoints lie in K. c is contained in an apartment A. Since the apartments 0Tit, A and K have antipodal regular points in common, they coincide as a consequence of lemma 3.6.1. A is unique by corollary 4.6.4. [] Lemma 4.6.7. -- Let A be an apartment in X. If c is a geodesic arriving at p ~ A, it can be extended into A. Proof. -- If ~ is the direction of c at p then, by lemma 3.10.2, B has an antipode in the spherical apartment Z~ A. Hence c has an extension into A. [] Corollary 4.6.8. -- For any point x and any apartment A in X the geodesic cone over A at x lies in the cone over OTlt~ A. In particular, it is contained in a finite union of apartments passing through x. RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 159 Sublemma 4.6.9. -- Let Y be a Euclidean building with associated admissible spherical polyhedron Amo a. Then for each direction ~ e int(Amoa) the subset 0-~(~) in the geometric boundary O~ Y is totally disconnected with respect to the cone topology. Pro@ -- Suppose that y,y',y" e Y so that O(yy')----O(yy")= b. Define the point z byyy' nyy" =yz. If z ~ey',y" then the angle rigidity axiom EB2 implies that /,(Y',Y") >t ~o :---- 2c'damoa(8, 0 Amoa) and by triangle comparison we obtain: [y' z t ~< sin ~o .d(y',yy"). As a consequence, for each z ~ Y the closed subset { ~ e 00o Y ] 0(4) -= 8 and z Ey~ } of 0- ~(~) is also open and we see that each point in 0- ~(8) has a neighborhood basis consisting of open and closed sets. ca 4.7. Subbuildlngs A subbuilding X' c X is by definition a metric subspace which admits a Euclidean building structure. This implies that X' is closed and convex and that 0T~t~ X' is a spherical subbuilding of O~ X which is closed with respect to the cone topology. We consider a partial converse: Proposition 4.7.1. -- Let X be a Euclidean building and B ~_ OTlt~ X a subbuilding of full rank. Then the union X' of all apartments A with 0Tit~ A _ B has the following properties: 9 If X" is closed then it is a subbuilding offuU rank and the subbuilding OT~ ~ X' ~ OTlt~ X is the closure B of B with respect to the cone topology. Furthermore, X" is the unique subbuilding with OTi ~ X' = B. 9 If X is discrete or locally compact then X' is closed. Proof. -- Observe that X' ~) { A apartment [ 0Tlt~ A ___ B } ---- u { A apartment [ 0~it~ A _c ~ }. We first show that X' is a convex subset. Consider points Xl, x2 e X'. There are apart- ments A~ with x 1 e A~ _c X'. By lemma 3.10.2, there exist ~e0Tl~A ~ with /__,4(x3_i, ~)= r:. The canonical map +:0Ti~X-->Z| is a building morphism and satisfies the assumption of proposition 3.11.2. Thus, since /,1(~1, ~) = 7:, there is an apartment 0~l ~ A _ X' which contains ~1, ~2 and projects isometrically to Z,~ X via +. This means that x 1 ~ A. Consequently x 1 x2 C A and X' is convex. Similarly, one shows that any ray and geodesic in X' lies in an apartment A which is limit of apart- ments A, with Ow~t, A,_ B, i.e. ~ A c B and A ___ X'. The building axioms are inherited from X and if X' is a closed subset then it is complete and a Hadamard space. This proves assertion (i). (ii) Assume that X is discrete and x E X'. Any point x' e X' lies in an apartment A _~ X', and if x' is sufficiently close to x then A contains x. Hence X' is closed in this case. 160 BRUCE KLEINER AND BERNHARD LEEB Assume now that X is locally compact and that (x,) C X' is Cauchy with limit x E X. Let p e X' be some base point. Any segment ~b~, lies in some apartment A, _~ X' and we can pick rays px~, 4, in A, so that lim x', = x and 04, = 0fi-~. After passing to a subsequence, we may assume that (~,) converges to a point 4 ~ B. Since 04, = 04, lemma 4.1.2 implies that the segments P4, n P4 C X' n P4 converge to P4. Hence P4 contains x and lies in X'. [] 4.8. Families of parallel flats Let X be a Euclidean building and F _ X a flat. If another flat F' has finite Hausdorffdistance from F then F and F' bound a flat strip, i.e. an isometrically embedded subset of the form F � I with a compact interval I C R. In this case, the flats F and F' are called parallel. Consider the union P~ of all flats parallel to F; PF is a closed convex subset of X and splits isometrically as P~F � Proposition 4.8.1. -- The set PF is a subbuilding of X and Y admits a Euclidean building structure. Proof. -- By proposition 4.6.1, PF is the union of all apartments which contain F in a tubular neighborhood, and 0Tl ~ PF is the union of all apartments in O~lt~ X which contain the sphere 0Tlt~ F. The subset 0Tlt~ PF -- 0Tit8 X is convex by lemma 4.1.2 and a subbuilding by proposition 3.10.3. Proposition 4.7.1 implies that Px, is a subbuil- ding of X. As a consequence, the Hadamard space Y inherits a Euclidean building structure. [] If dim(F) ----rank(X) -- 1, then Y is a building of rank one, i.e. a metric tree. Since Eu Y is in this case a zero-dimensional spherical building, any two raysy~l andy-~ in Y either initially coincide or their union is a geodesic. This implies: Lemma 4.8.2. -- (i) Let H1 and H~ be two flat half-spaces of dimension rank(X) whose intersection HI c~ H~ coincides with their boundary flats. Then H I u H 2 is an apartment. (ii) If Aa, A~, A 3 ~_ X are apartments, and for each i oe j the intersection A~ c~ Aj is a half-apartment, then A 1 c~ A~ n A3 is a wall in X. Lemma 4.8.3. -- Let CI, C~, C3C 0~lt~X be distinct adjacent chambers, with = C1 n C~ n C a their common panel. Then there is a p ~ X so that if Cone(p, ~) = w{p~ [ 4 ~}, then log~,(C,) C Z~, X are distinct chambers for every p' ~ Cone(p, n) and any apartment A C X such that OTis, A contains two of the C, must intersect Cone(p, n). Proof. -- Let m C 0Tlt~ X be a wall containing the panel 7:. Then each chamber C~ lies in a unique half-apartment h, bounded by m, and pairs of these half-apartments RIGIDITY OF OUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 161 form apartments. Let A~ be the apartment in X with 0~l~A~----h~ u hi. By lemma 4.8.2, ['lAij is a wall MCX, and we clearly have 0~i~M=m. IfpeM, then the half-apartments log~ h~ C Z~ X are bounded by log~ m = Z~ M, so they are distinct; otherwise [1 A~ :~ M. Hence the chambers log~ C~C log~h~ are distinct chambers. If A C X is an apartment with C~ u Cj C 0~ A, i + j, then there are chambers (~, C~ C A n A~5 with 0Ttt, C~ = C~, 0~ Cj = C~.. The Tits boundary of the Weyl poly- hedron P ---- A~ n A contains C~ u Cj, so it intersects Cone(p, n). [] 4.9. Reducing to a thick Euclidean building structure This subsection is the Euclidean analog of section 3.7. Definition 4.9.1. -- Let X be a Euclidean building modelled on the Euclidean Coxeter complex (E, W,ff), with atlas d'. The affine Weyl group may be reduced to a reflection subgroup W~'ff C War f /f there is a W'~f compatible subset d' C ~r forming an atlas for a Euclidean building modelled on (E, W~). In contrast to the spherical building case, the affine Weyl group of a Euclidean building does not necessarily have a canonical reduction with respect to which it becomes thick. For example, a metric tree with variable edge lengths does not admit a thick Euclidean building structure. However, there is always a canonical minimal reduction, and this is thick when it has no tree factors. Proposition 4.9.2. -- Let X be a Euclidean building modelled on (E, W~). Then there is a unique minimal reduction W'af C W~x f so that (X, E, W~ff) splits as a product 1-I Xi where each X~ is either a thick irreducible Euclidean building or a 1-dimensional Euclidean building. The thick irreducible factors are either metric cones over their Tits boundary (when the affine Weyl group has a fixed point) or the# affine Weyl group is cocompact. Proof. -- We first treat the case when (0~t~ X, Am~l) is a thick irreducible spherical building of dimension at least 1. Step 1. -- Each apartment A C X has a canonical affine Weyl group G x. If A C X is an apartment, a wall M C A is strongly singular if there is an apartment A' s X so that A n A' is a half apartment bounded by M. Since 0~lt, X is thick and irreducible, for every wall m C 0~l ~ A there is a strongly singular wall M C A with 0~ M -~ m. Sublemma 4.9.3. ~ The collection ~r of strongly singular walls in A is invariant under reflection in any strongly singular wall in A. Proof. ~ Note that a wall M C A is strongly singular if and only if Y~ M C Y,~ X is a wall with respect to the thick building structure (Y~ X, A~oa(p)); this is because any half-apartment h C Y,~ X with boundary Zp M can be lifted to a half-apartment 21 162 BRUCE KLEINER AND BERNHARD LEEB H C X with boundary M, Y,, H = h by applying proposition 3.11.4 to the surjective spherical building morphism log~ : 0xlt~ X --> 1~ X. If M 1, M, C A are strongly singular walls intersecting at p ~ A, then ~ M~ is a A~o~(p) wall in Z~ A C Z~ X, and so if we reflect Z~ M, in Z, M~ (inside the apart- ment Z, A), we get another A~o~(p) wall which is then the space of directions of the desired strongly singular wall M~. Now suppose that M1, Mz ~ ~r are parallel. Amo ~ is irreducible so there is a strongly singular wall Ma intersecting both M~ at an acute angle. Reflect M, in M a to get M,, reflect Ma in M 1 to get M~, and M, in M~ to get M6, and finally reflect M~ in M~ to get a wall which is the image of Mz under reflection in M~. The walls M~ are all in ~g~x, so we are done. [] Proof of Proposition 4.9.2 continued. -- Hence for every apartment A C X the collection of strongly singular walls in A gives us a group G A C Isom(A) which is gene- rated by reflections. Step 2. -- The group Ga is independent of A. Since 0TI . GAC Isom(~ A) is an irreducible Coxeter group, it follows that G A is either a discrete group of isometrles or it has a dense orbit. When G A is discrete, it is generated by the reflections in the strongly singular walls which intersect a given Gx-chamber in codimension 1 faces. When G x has a dense orbit, it is generated by all the reflections in strongly singular walls passing through any open set. If two apartments A x and Az intersect in an open set, it follows that Gxt is isomorphic to GA,; therefore G A is independent of A. So there is a well-defined Coxeter complex (E, W~) attached to X. Step 3. -- Finding (E, W',n) apartment charts. If Z is a convex domain in an apart- ment A C X and t : U --> Z is an isometry of an open set U C E onto an open set in Z, then there is a unique extension of ~ to an isometry of a convex set Z C E onto Z. Pick an apartment A 0 C X and an isometry t o : E ~ A 0 which carries W~C Isom(E) to GA0. Then restrict to a W~ff chamber ~1o C E and its image C0 d~ ~o(Co) C A o. Given any chamber (3 C X, there is an apartment A x containing subchambers of C and C 0. There is a unique isometry t 1 : E -+ A x so that t~-a and t o 1 agree on the subchambers Co n A 0 C Ax, and a unique isometry t o : E ~ C ~ C so that t~ x and q- 1 agree on the subchamber C n A x. If A, is another apartment with 0T~ ~ Co, 0T~,~ C C 0Tit, A,, we get another isometry t,:E ~A,; but the convex set A~ c~ A 2 contains subchambers of Co and C, so ~- ~ and t~-1 agree on a subchamber of C. Therefore t c is independent of the choice of apartment asymptotic to Co u C. Sublemma 4.9.4. -- Let A C X be an apartment, and let Ca, C2 C 0Tl~, A be adjacent Amod-chambers ((I 1 n C2 is a panel). For i = 1, 2 we let tel(A):E ~A be the unique -- t -- isometric extension of tUi where C~C A is a W~-chamber with O~l ~ C~-~ Ci. Then o tol(a) RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 163 Proof. -- For i = 1, 2 let A,~ C X be an apartment with C O t3 C, C 0~t, A,. If C t is contained in the convex hull of C O ~ Cn (or C n C ConvexHull(C 0 ~ C~)) then C t t3 C,C 0~l~(A c~ An) , so the sublemma follows from the fact that ~(A) restricted to A nA n coincides with '~-alxnx," So we may assume that there is a chamber CsC 0~I~A1 tn 0T~A n which meets C~ and C~ in the panel ~----C~ c~ C~. By lemma 4.8.3 (applied to the original Euclidean building (X,E, W~)), there is a pointp e At c~ A n so that Cone(p, ~) C A~ n A n and log~(C~) C Z~ X are distinct chambers for i = 1, 2, 3. Therefore ~i -1 and t~-a agree on Cone(p, ~). Hence the isometrics ~c a(A), ~-1 agree on Cone(p, ~), which means that ~c~(A)o ~c~(A):E-+E is a reflection. But since Z~(Cone(p, r~)) = log~(Ct) tn log~(Cn) tn log~(Ca), Cone(p, n) spans a strongly singular wall in A and so the reflection ~ca(A) o :c~(A) ~ W~. [] Proof of Proposition 4.9.2 continued. -- By sublemma 4.9.4, we see that for each apartment A C X, there is a canonical collection of isometrics ~: E-+ A which axe mutally W~ compatible, and which are compatible with the ~c:C--+ C for every chamber C C A. We refer to such isometrics as W~-charts, and to the collection of W~-charts (for all apartments) as the (E, W~) atlas ~r Sublemma 4.9.5. -- Let A1, A n C X be apartments with d-dimensional intersection P = A~ n A~. Ifp ~ P is an interior point of the Weyl polyhedron P, then there is an apartment As C X so that Aa contains a neighborhood of p ~ P, and A s tn A~ contains a Weyl chamber. Proof. -- We have Y~ A 1 c~ Y~ A 2 = Y~ P by lemma 4.4.3. Let ax C Y.~ P be a d -- 1-dimensional face of Y~ P, and let ~ be the opposite face in Y~ P. If ~1 C Y~ A 1 is a chamber containing ~1, then we may find an opposite chamber ~z C Y~ A n. But then *n contains a face opposite ~i, and this must be an since each face in an apartment has a unique opposite face in that apartment. Let C, C 0Tt ~ A~ be the chamber such that log~ C, = v,. Then there is a unique apartment A s C X with Ct u C n C 0~ A s. Y,= PC ~I~ As, so A s has the properties claimed. [] Proof of Proposition 4.9.2 continued. -- If A1, A n C X are apartments with A1 tn A n ~e ~, then any W~ charts ~,:E ~ A, axe W~ compatible since by sublemma 4.9.5 we have a third apartment A s C X so that ~t and ~ are both W~ compatible with ts : E -+ A s on an open set U C A 1 n A 2. Hence .~' gives X the structure of a Euclidean building modelled on (E, W~). From the construction of W~f it is clear that (X, a~r is thick. Step g. -- The case when X is a 1-dimensional Euclidean building, i.e. a metric tree. Let A 0c X be an apartment, 0Tl ~A 0 ---- { ~x, ~n}" For each p eX let ~x0(P) eA o be the nearest point in Ao, and PA0 ~A0 be a point (there are at most two) with d(PA,, ~*0(P)) ----- d(p, A0). Let ~gf C A o be the set of points PAo where p E X is a branch point: [ l~ X [ >/3; let G C Isom(A0) be the group generated by reflections at points in~r For each ~ E 0~l ~ X\~ x there is a unique isometry t~ from the apartment A o -= ~x ~2 to the apartment ~1 ~ which is the identity on the half-apartment ~1 ~n tn ~x ~. If~a ~e ~9, 164 BRUCE KLEINER AND BERNHARD LEEB then we have two isometries ~1, t2 : A0 -+ 41 42 where ~-1 agrees with ~i on B1 4, r 41 42- By inspection t~- 1 o h ~ G. Hence for each apartment A C X we have a well-defined set of isometrics A 0 ~ A. As in step 3 it follows that these isometrics are G-compatible, so they define an atlas d' for a Euclidean building structure on X. Step 5. -- X is an arbitrary Euclidean building modelled on (E, War ). Let Waa= 0Tl~, W~, and let W' C W C Isom(0~lt, E) be the canonical reduced Weyl group of 0Tlt~ X given by section 3.7. Let W,aC Isom(E) be the inverse image of W' under the canonical homomorphism Isom(E) -+ Isom(0~l ~ E). Let 0' : 0~lt~ X ~ A~o a a~ S/W' be the A~oa-anisotropy map. We may define A~oa-directions for rays x~ C X by the formula -- t ! 9 9 0'(x4) -= 0'(4) e ZX~o a. We define the ZX~oa-directaon of a geodesic segment ~ C X by setting 0'(~) = 0'(x~l) for any ray x~ 1 extending xy; if x~2 is another ray extending then ~1 E 0Tlta X and ~2 e 0~1 ~ X are both antipodes of ~ e 0~l u X where ~ is a ray extending~, so 0'(~) is well-defined. The remaining Euclidean building axioms follow easily from the fact that any two segments ~-~, ~ initially lie in an apartment A C X (corollary 4.4.3) and for our compatible (E, War) apartment charts we may take all isometric embeddings i : E -~ X for which O~t~ i : 0~1 ~ E ~ O~ X is an apartment chart for (OTl~, X, A~oa). We may now apply proposition 4.3.1 to see that (X, E, W,a) splits as a product of Euclidean buildings (X, E, W~f) = (IIx~, HE~, II W,sf) ' so that each 0~,t~ X i is irreducible. Let (W~r)' C War, off, be the canonical subgroup and atlas constructed in steps 1-4, and set War' = II (W~)' C Isom(E), off' = Hoff,. Then (X~., E~, (War),~ 'off,) has the properties claimed in the proposition. Fix an apartment A 0 C X and a chart ~ao s off. If Ao, ..., A~ = A 0 is a sequence of apartments so that A~._x n A~. is a half- apartment for each i, then there is a unique isometry g~ : A~_ ~ ~ A, so that g~ is the identity on Ai _ ~ n A i. Axiom EB4 implies that g~ o ... o g~ o ~0 e off for each i, so in particular g = g~ o ... o gx ~ ta0,(Wae) 9 From the construction of (Wi~) ' it is clear that the group of all such isometrics g : A 0 -+ A 0 contains tko(W'~) C Isom(A0) where ~o e off" So War C W~ is a minimal reduction of W~. [] 4.t0. Euclidean buildings with Moufang bound~T This is a continuation of section 3.12. Proposition 4.10.1 (More properties of root groups). -- Let B be a thick irreducible spherical building of dimension at least 1, and let X be a Euclidean building with Tits boundary B. 1. For every root group U, C Aut(B, Amoa) and every g ~ U, there is a unique automorphism gx : X ~ X so that OT~ ~ gx = g" In other words, if G is the group generated by the root groups, then the action of G on OT~, X " extends " to an action on X by building automorphisms. Henceforth we will use the same notation to denote this extended action. 2. Suppose g ~ U~ is nontrivial. IrA c_ X is an apartment such that O~it, A ~ a, then g(A) c~ A is a half-apartment; moreover Fix(g) n A = g(A) ~ A. RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 165 Proof. -- See [Ron, Affine buildings II, esp. prop. 10.8], or [Ti2, p. 168]. For the remainder of this section X will be a thick, nonflat irreducible Euclidean building of rank i> 2. Therefore Amo a is a spherical simplex with diameter < ~/2 and the faces of 0Tlt~ X define a simplicial complex. Lemma 4.10.2. -- Let A C X be an apartment, Poe X, p ~ A the nearest point in A, and a C OA a root. Then the stabilizer of Po in the root group U, fixes p. Proof. -- Using lemma 3.10.2 extend the geodesic segment PoP to a geodesic rayP0 ~ ----= PoP up~ so that the rayp~ lies in the half apartment Cone(p, a) C A. Ifg e U~ fixes P0, then it fixes the ray P0 ~, and hence the half-apartment Cone(p, a). [] We now assume that the spherical building (0~lt, X , Amoa) is Moufang. Pick p e X, and let (Y,~ X, A~oa(p) ) denote the thick spherical building defined by the space of directions Z~ X with its reduced Weyl group (see section 3.7). Suppose H+ C X is a half-apartment whose boundary wall passes through p, h+ asj ~ H+ C Z~ X is a Z~oa(p) root, and let a+ = 0~lt, H+ C 0~t, X. If U,+ is the root group associated to a+, and Va+C Ua+ is the subgroup fixing p, then we have a homomorphism Z~ : V,+ -+ Aut(Y,~ X, A~oa(p) ). Lemma 4. t0.3. -- The image of V,+ is the root group Uh+ associated with h+ , and this acts transitively on apartments in Z~ X containing h+. In particular, ('Z~ X, A~oa(p) ) is a thick Moufang spherical building. Proof. -- By corollary 3.11.5, if h_ C Z~ X is a A~oa(p) root with Oh_ = Oh+ = Z~(0H+), then there is a half-apartment H_ C X so that H_ and H+ have the same boundary and Z~ H_ = h_. Given two such Amoa(p) roots ha., h~ C Z~ X so that h ~_ u h+ forms an apartment in Z~ X, we get two half apartments H ~_ so that H ~_ u H+ forms an apart- ment in X. Since (0T~ u X, Amoa) is Moufang, the root group UA§ C Aut(0T~ u X, A~a) contains an element which carries H a_ to H~_. By 3.12.2, g " extends " uniquely to an isometry g : X ---> X which carries the apartment Ha. u H+ to the apartment H 2 _ u H+, fixing H+ (see 4.10.1). It remains only to show that the isometry Z~g : Z~ X -+ lg~ X is contained in the root group Ua+C Aut(Z~ X, A~oa(p) ). Clearly Z~g fixes h+. Let C C Z~ X be a A~oa(p) chamber such that C n h+ contains a panel ~ with r~ r Oh+. Using proposition 3.11.4 we may lift C to a (subcomplex) C C 0~,~ X so that C n Oa+ maps isometrically to C c~ Ok+ under log~ : 0~t~ X-+ ~ X. g fixes an interior point of C, so Y~ g fixes an interior point of C, which implies that Y~ g fixes C as desired. [] Definition 4.10.4. ~ A point s ~ X is a spot/f either 1. The affine Weyl group War has a dense orbit or 2. Wm is discrete and s corresponds to a O-simplex in the complex associated with X. If A _ X, then Spot(A) is the set of spots in A. 166 BRUCE KLEINER AND BERNHARD LEEB Lemma 4.t0.5. -- If A C X is an apartment, Po ~ A is a spot, then for every p 4= Po there is a root a C O~t~ A and a g ~ U~ so that g fixes Po but not p. Proof. -- For each Amoa(po t~ ) root h+ C X~o X we have a singular half-apartment H+ C A with X~o H+-= h+, and this gives us a root a+--= 0Ti ~ H+ C O~l ~ X, the root group Ua. , and the subgroup Va+ C Ua+ fixing Po. By lemma 4.10.3, the image of Va. in Aut(X~oX , A~oa(Po)) is the root group Uh.. Since (E~oX , A~oa(Po)) is Moufang, the group G~o generated by the Vh+'s as h+ runs over all th Amoa(Po ) roots in X~ A acts transitively on A~oa(P0 ) chambers in X~oX (see 3.12.2). If p e X--Po is fixed by every Va§ , then PoP E E,0 X is fixed by G~o , which means that it lies in every A~oa(P0 ) chamber of E~o X, forcing Po~ e X~o A. Hence the point q ~ A nearest p is different from Po, so we may find a singular half-apartment H+ C A containing Po but not q (because Po is a spot), and use the root group UoTitsm to move q while fixing H+. This contradicts the assumption that p is fixed by every V,§ [] Proposition 4.10.6. -- Let X be a thick, nonflat Euclidean building of rank at least two, and suppose OTl ~ X is an irreducible Moufang spherical building. Let G C Aut(0Tl ~ X, Amod) be the subgroup generated by the root groups of O~it~ X, and consider the isometric action of G on X. 1. The fixed point set of a maximal bounded subgroup M C G is a spot, and the stabilizer of a spot is a maximal bounded subgroup. 2. A spot p ~ X lies in the apartment A C X if and only if p is the unique spot in X which is fixed by the stabilizer of p in U, for every root a C OTlt~ A. 3. If A C X is an apartment, and a C O~it~ A is a root, then as g runs through all non-trivial elements of U~, we obtain all singular half-apartments H s A with OTTO, H = a as subsets A n Fix(g). Proof. -- Let M _ G be a maximal bounded subgroup. By the Bruhat-Tits fixed point theorem [BT], M has a nonempty fixed-point set Fix(M), which contains a spot since when W~ is discrete the fixed point set of a group of building automorphisms is a subcomplex. By lemma 4.10.5, we see that if p0 ~ Fix(M), then maximality of M forces Fix(M) = {P0}. Conversely, if p0 ~ X is a spot, then the stabilizer of p0 has fixed point set {P0} by lemma 4.10.5, and by the Bruhat-Tits fixed point theorem, the stabilizer is a maximal bounded subgroup. For every p ~ X and every apartment A C X, let G(p, A) be the group generated by the stabilizers of p in the root groups U,, where a C 0~ A is a root. If p ~ A C X is a spot, then by lemma 4.10.5 we have Fix(G(p, A)) = {p}. Ifp CAC X, then the nearest point p0 ~ A top is contained in FIX(G(p, A)) by lemma 4.10.2; hence Fix(G(p, A)) contains a spot other than P0. Claim 3 follows from property 2 of proposition 4.10.1, the fact that 0~ X is Moufang, and the fact that every singular half-apartment is the intersection of two apartments. [] RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 167 Definition 4.10.7. -- If A C X is an apartment, then the half-apartment topology on Spot(A) is the topology generated by open singular half-apartments contained in A. With the half-apartment topology, Spot(A) is discrete when War is discrete and coincides with the metric topology when War has dense orbits. 5. ASYMPTOTIC CONES OF SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS In this section we arrive at the heart of the geometric part in the proof of our main results. We show that asymptotic cones of symmetric spaces and ultralimits of sequences of Euclidean buildings (of bounded rank) are Euclidean buildings. Our main motivation for choosing the Euclidean building axiomatisation EB1-4 is that these axioms behave well with respect to ultralimits. Indeed, the Euclidean building axioms EB1, EB3 and EB4 which are also satisfied by symmetric spaces, i.e. the existence of &moo-directions and an apartment atlas, pass directly to ultralimits. However, unlike Euclidean buildings, symmetric spaces do not satisfy the angle rigidity axiom EB2. The verification of EB2 for ultralimits of symmetric spaces (lemma 5.2.2) is the only technical point and, as opposed to the building case (lemma 5.1.2), non-trivial. Sym- metric spaces satisfy angle rigidity merely at infinity; their Tits boundaries are spherical buildings. Intuitively speaking, the rescaling process involved in forming ultralimits pulls the spherical building structure (the missing angle rigidity property) from infinity to the spaces of directions. 5.1. Ultrallmlts of Euclidean buildings are Euclidean buildings Theorem 5.1.1. -- Let X,, n ~ N, be Euclidean buildings with the same aniso- tropy polyhedron A=oo. Then, for any sequence of basepoints ,, ~X,, the ultralimit (Xo, ,~) = r ,,) admits a Euclidean building structure with anisotropy polyhedron Amo o. Proof. ~ The building Xo is a Hadamard space (lemma 2.4.4). A Euclidean building structure on Xo consists of an assignment of Amoo-directions for segments (axioms EB1 + EB2) and of an atlas of compatible charts for apartments (axioms EB3 + EB4), cf. section 4.1.2. We assume that X has no Euclidean de R_ham factor. The general case allowing a Euclidean de Rham factor is a trivial consequence. EB1. -- We can assign a Amoo-direction to an oriented geodesic segment in X,~ as follows. A segment xoyo arises as ultralimit of a sequence of segments x,y, in X, and we define the direction as: (37) 0(~) := ~o-lim 0(x-~y~) ~ Amo d. The ultralimit (37) exists because Amo d is compact. Inequality (28) in EB1 passes to the ultralimit: damoa(o~-lim 0(x--~-~.), o~-lim 0(x-~.)) ~< ~_~(y~, z,o ). 168 BRUCE KLEINER AND BERNHARD LEEB This implies that the left-hand side of (37) is well-defined and damoa(0(x-~--y~), 0(~)) ~< 2,o(yo, Zo). Thus axiom EB1 holds. It implies lemma 4.1.1. Therefore, segments which contain a given segment have the same Amod-direction and we can assign Amoa-directions to geodesic rays. EBb. -- Since geodesics are extendible in Xo, it suffices to show: Lemma 5.1.2. -- If Xo ~ Xo and to, ~o ~ O~it~ Xo then /~o(~o, ~o) is contained in D := D(0(x o ~o), 0(x-~)). Proof. -- The rays x o to and x o ~o are ultralimits of sequences of rays x s ~s and x s~. in X s and we can choose ~s,~,EOTit~X~ SO that 0(~s)=0(x oto) and 0(~1.) = 0(x--~-~). Let Ps : [0, oo) -+ K s be a unit speed parametrisation for the geodesic ray x, ~,. The angle /p,m(~,, ~,) is non-decreasing and continuous from the right in t (lemma 2.1.5) and, since X, satisfies EB2, takes values in the finite set D. For d e D set ts(d ) := min{ t >/0 :/p.m(~s, ~,) >/ d} ~ [0, oo] and to(d) := c0-1im ts(d ). Then there exist d 0eD and T>0 with to(do)----0 and 2T~< to(d ) for all d>d o . The t t points x', := ps(ts(d0)) and x'~" := ps(T) satisfy for o~-all n, x o := c0-1im x s = xo, x' o' := r x': 4: xo and the ideal triangle A(x',, x'~', ~) has angle sum ~. By a version of the Triangle Filling Lemma 2.1.4 for ideal triangles in Hadamard spaces, A(x's, x',', ~s) can be filled in by a semi-infinite flat strip S s. The ultralimit co-lim S, is a semi-infinite flat strip filling in the ideal triangle A(xo, x'~', ~o) and therefore L,o(~o, ~o) = c0-1im/,~(~s, ~s) = do e D, as desired. EB3 and EBg. -- After enlarging the affine Weyl groups of the model Coxeter complexes of the buildings Xs, we may assume that the X s are modelled on the same Euclidean Coxeter complex (E, W~ff) whose affine Weyl group W,s ~ contains the full translation subgroup of Isom(E), i.e. p-l(W) = War f where p : Isom(E) -+ Isom(0~i m E) is the canonical homomorphism (26) associating to an affine isometry its rotational part. (Here we use that the X, do not have Euclidean factors.) The atlases M s for the building structures on X s give rise to an atlas for a building structure on X o as follows: If ts e~'s are charts for apartments in X s so that co-limd(t,(e), *s) < oo for (one and hence) each point e E E, then the ultralimit %, := o~-lim t s : E -+ X,o is an isometric embedding which parametrises a flat in X o. The collection d o of all such embeddings ~o satisfies axiom EB3 in view of lemma 2.4.4. Axiom EB4 holds trivially, because coordinate changes ~2 x o t~, between charts to, ~ e d o are A~oa-direction-preserving isometrics between convex subsets of E and such isometrics are induced by isometrics in p-I(W) = W~u. Hence d o is RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 169 an atlas for a Euclidean building structure on X,~ with model Coxeter complex (E, Waff) , and the proof of the theorem is complete. [] Corollary 5.1.3. -- Let X be a Euclidean building modelled on the Coxeter complex (E, W~) and denote by ~V~ the subgroup of Isom(E) generated by W,~ and all translations which preserve the de Rham decomposition of (E, W~) and act trivially on the Euclidean de Rkam factor. Then any asymptotic cone X~ inherits a Euclidean building structure modelled on (E, ~r The building X,o is thick if X is thick and the affine Weyl group W~ is cocompact. Proof. -- We have X,o = c0-tim(X,, .,) where the X, are scale factors with co-lim ~. = 0, X, is the rescaled building ~, X, and., ~ X, are base points; X,o inherits the Euclidean building structure modelled on (E, "~r~) which was constructed in the proof of the previous theorem. Suppose now in addition that X is thick and Wm is cocompact. Then any wall w, C X. branches, i.e. there are half-apartments H,~ C X,, i = 1, 2, 3, so that the inter- section of any two of them equals w, and the union of any two of them is an apartment (lemma 4.8.2). If a sequence of walls w, satisfies co-lim d(w,, ,,)< oo, it follows that the ultralimit of the sequence (w.) is a branching wall in X~. Since W~ is cocompact by assumption, there is a positive number d so that any flat in X, whose ideal boundary is a wall in O~t ~ X, lies within distance at most d from a branching wall in X. In view of co-lim ~, = 0, this implies that any flat in X,o , whose ideal boundary is a wall in 0~i~ X,~, is a branching wall. Thus, the Euclidean building structure on X,~ is thick. [] 5.2. Asymptotic cones of symmetric spaces are Euclidean buildings We start by recalling some well-known facts from the geometry of symmetric spaces which will be needed later; as references for this material may serve [BGS, Eb]. Let X be a symmetric space of noncompact type. In particular, X is a Hadamard manifold, i.e. a complete simply-connected Riemannian manifold of nonpositive sectional curvature. To simplify language, we assume that X has no Euclidean factor. The identity component G of the isometry group of X is a semisimple Lie group and acts transitively on X. A k-flat in X is a totally geodesic submanifold isometric to Euclidean k-space. We recall that G acts transitively on the family of maximal flats. In particular, any two maximal flats in X have the same dimension r; it is called the rank of X. We will call the maximal flats also apartments. Pick an apartment E in X and let W~ be the quotient of the set-wise stabiliser Staba(E ) by the pointwise stabiliser Fixa(E ). Then W~ can be identified with a subgroup "of Isom(E). This subgroup is generated by reflections at hyperplanes and contains the full translation group. We call (E, W~) the Euclidean Coxeter complex associated to X. Its isomorphism type does not depend on the choice of E, because G acts transitively on apartments. Consider the collection of all isometric embeddings L : E -~ X so that W~r is identified with Stabo(~(E))/Norn~(~(E)). Walls, singular flats, Weyl chambers et cetera are defined 22 170 BRUCE KLEINER AND BERNHARD LEEB as images of corresponding objects in E via the maps ,. Note that the singular flats are precisely the intersections of apartments. The induced isometric embeddings 0Tl ~ ~:0~lt~ E-+ 0Tlt~ X form an atlas for a thick spherical building structure on 0~l ~ X modelled on the spherical Coxeter complex (0~t~E, W)= 0~t~(E, W~). The group W is isomorphic to the Weyl group of the symmetric space X. Composing the anisotropy map 0OTit, X : OTtU X -+ Amo d with the map SX -+ 0zi~ X which assigns to every unit vector v the ideal endpoint of the geodesic ray t ~-~ exp(tv) one obtains a natu- ral map (38) 0 : SX -+/Xmo d from the unit sphere bundle of X to the anisotropy polyhedron boa- We will call 0(v) the A~oa-direction of v ~ SX; Amos-directions of oriented segments, rays and geodesics are defined as the boa-direction of the velocity vectors for a unit speed parametrisation. The orbits for the natural G-action on SX are precisely the inverse images under 0 of points. Let S~ X be the unit sphere at p ~ X, equipped with the angular metric, and let Gr be the isotropy group of p. Then 0 induces a canonical isometry SflG~ -~ Amo a where S~/Gt is equipped with the orbital distance metric. The quotient map Sv X ~ Amo d is 1-Lipschitz and, for any x,y E X we have the following counterpart to inequality (28): (39) d~m~(O(p-k), 0(p~)) ~< A~(x,y) ~< ~-~(x,y). The goal of this section is to prove the following theorem. Theorem 5.2.1. -- Let X be a non-empty symmetric space with associated Euclidean Coxeter complex (E, W~rr). Then, for any sequence of base points ,, ~ X and scale factors ~ with co-lira ?,, = 0, the asymptotic cone X,o = o~-lim(~, X, ,,) is a thick Euclidean building modelled on (E, W~). Moreover, X,o is homogeneous, i.e. has transitive isometry group. Proof. --EBL- Let Amo a be the anisotropy polyhedron for (E, W~). The construction of A~od-directions for segments in X~ is the same as in the building case. We define directions by (37) and (39) implies that the definition is good and that EB1 holds. EBB and EB4. -- The Euclidean Coxeter complex (E, W~) is invariant under rescaling, because W~rC Isom(E) contains all translations. Apartments in X~ and their charts arise as ultralimits of sequences of apartments and charts in X, and axioms EB3 and EB4 follow as in the building case, cf. section 5.1. EB2. -- The only nontrivial task is to verify the angle rigidity axiom EB2. This will be done in the following Iemma. Lemma 5.a.2. -- If p ~ Xo, and Xx, x~ ~ X,,, -- {p}, then s x2) E D(0(ffX~I), 0(~-22)). Proof. -- If z~ ~ffx-~l -- P and z~ -->-p, then L,~(xa, x~) --->- s x~) and Z_,,~ (p, x~) ~ ~r --/_~,(xt, x~) by lemma 2.1.5. Since 0 (z~ xz) ~ 0 (fi~) we can find x~, ~ z~ xx, RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 171 I I 'I ~ p t ~ I I x~, ~ z, x2, and Pl, ~ z,p such that/,~(xlk , x~) -+/~(xl, x2) ,/,~(Pk, x2k) -+ ~ -- /~(xl, x~), and O(z~x~,)= O(z'kxz)~0(p--~2). Since geodesic segments in X,o are ultralimits of geodesic segments in X, X, we can find sequences p,, x~k , x~, z, e X such that z, Epk Xl, , -+ x,), - x,), and finally [z,x~,[, [z~xz~[, [zkp,] ~oo. Applying a sequence of elements g~ e G = (Isom(X)) ~ we may assume in addition that z, is a constant sequence, z k - o. Hence the sequences of segments oxl, , ox~, opk subconverge to rays o~1, o~, and o'~ respectively, which satisfy the following properties: 1. 00Tit, X(~ ) = 0(0~t ) = 0(~); 2. /T,t~(~, ~*) ~< /~(X~, X,), /T,~(~, ~.) <~ ~ -- /--~(X~, XZ) by lemma 2.3.1 ; 3. o~ 1 u ~-~ is a geodesic, so /~lt,(~a, ~) ---- :~. We conclude that /_~(xx, xz) = /~,~(~x, ~,) ~ D(0(~a), 0(~2)) = D(0(P-~a), 0(ffk~,)) as desired. [] Hence we have constructed a Euclidean building structure on Xo. Since G acts transitively on Weyl chambers in X, it follows that the isometry group of Xo acts transitively on Weyl chambers in Xo; in particular, X o is homogeneous. To see that the building structure on X o is thick it is therefore enough to check that the induced spherical building structure of Z, o X,o modelled on (0~it, E , W) is thick. One way to see this is to construct a canonical isometric embedding ~ of the thick spherical building 0~t,X modelled on (0TIt. E , W) into Y.., X,o by assigning to ~ ~0~it~X the initial direction in ~ of the geodesic ray ~o-lim ,, ~ in X,0. That a is isometric follows, for instance, from the definition (8) of the Tits distance. This finishes the proof of the theorem. [] 6. THE TOPOLOGY OF EUCLIDEAN BUILDINGS In this section, X will denote a rank r Euclidean building. The main goal of the section is to understand homeomorphisms of X. As motivation for the approach taken here, consider a closed interval I topologically embedded in an R-tree T. Because every interior point p z I -- 0I of the interval disconnects T, every path c : [0, 1] -+ T joining the endpoints of I must pass thxough p, i.e. r I]) ~_ I. A similar phenomenon occurs in X if we consider topological embeddings of closed balls B C X of dimension equal to rank(X): if [c] e H,(X, 0B) and [ac] e H,_I(aB ) is the fundamental class of eB, then the image of the chain c contains B. By using 4.6.8, we can construct such c so that Image(c) -- U is contained in finitely many flats, where U is any given neighborhood of 0B. It follows that any b e B -- 0B has a neighborhood V b in X such that B c~ V b is contained in finitely many flats. 172 BRUCE KLEINER AND BERNHARD LEEB 6.1. Straightening simpHces If Z is a Hadamard space, there is a natural way to " straighten" singular simplices e : A~ -+ Z (cf. [Thu]). Using the usual ordering on the vertices of the standard simplex, we define the straightened simplex Str(,) by " coning ": if Str(e[%_l ) has been defined, then Str(e) is fixed by the requirement that on each segment joining p e A k_ 1 with the vertex opposite Ak_ 1 in A~, Str(,) restricts to a constant speed geodesic. The simplex Str(,) lies in the convex hull of the vertices of ,. This straightening operation induces a chain equivalence on C.(Z). By using the geodesic homotopy between Str(,) and ,, one constructs a chain homotopy H from the chain map Str to the identity with the property that Image(H(,)) ___ ConvexHull(Image(a)) for any singular simplex ,. When Z is the Euclidean building X, then it follows from lemma 4.6.8 that for every singular chain c e C~(Cone(X)), Image(Str(c)) is contained in finitely many apartments. Corollary 6.1.1. -- If Vc_ U=_X are open sets, then Hk(U ,V) =0 for every k > r = rank(X). Proof. -- If [c] H (U, V), then after barycentrically subdividing if necessary, we may assume that the convex hull of every singular simplex in c (respectively Oc) lies in U (respectively V). The straightened chain Str(c) determines the same relative class as c since Image(H(c)) C U, Image(H(0c)) C V and Str(c) - c = ~H(c) + H(~c). But the straightened chain is carried by a finite union of apartments (corollary 4.6.8), which is a polyhedron of dimension rank(X), so [Str(c)] = [c] = 0. [] Lemma 6.1.2. -- Let Z be a regular topological space, and assume that Hk(U1, Ua) = 0 for every pair of open subsets U, ~ U1 =- Z, k > r. If Y =_ Z is a closed neighborhood retract and U C Z is open, then the homomorphism H,(Y, Y n U) ~ H,(Z, U) induced by the inclusion is a monomorphism. In particular, the inclusion Y ~ Z induces a monomorphism H,(Y, Y --y) -+ H,(Z, Z -- y) of local homology groups for every y e Y. Proof. -- If [q] E H,(Y, Y n U), then there is a compact pair (K1, K2) _c (y, y n U) and [c2] e H,(K1, K2) so that i,([cs] ) = [q] where i : (K1, Ks) -+ (Y, Y n U) is the inclusion. If [q] is in the kernel of H,(Y, Y n U) -+ H,(Z, U) then there is a compact pair (KI, Ks) _c (K3, Ka) _c (Z, U) such that j, ( [c2] ) = 0, wherej : (K~, K2) ~ (K3, K4) is the inclusion. Let r:V-+ Y be a retraction, where V is an open neighborhood of Y in Z. Choose disjoint open sets W1, W2 C Z such that Y -- U _~ Wx, K 4 _ Ws; this is possible since Y -- U is closed, K 4 is compact, and Z is regular. Shrink V if necessary so that r-l(Y -- U)C W 1. We now have: H,(Y, Y n U) -+ H,(V, r-l(Y c~ U)) is a monomor- RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 173 phism since r is a retraction; H,(V, r-l(y n U)) --> H,(V u W2, r-~(Y n U) u W2) is an isomorphism by excision; H,(V u W2, r-l(Y n U) u W~) -+ H,(Z, r-~(Y n U) u W2) is a monomorphism by the exact sequence of the triple (Z, V to W2, r-~(Y n U) u W2) and H,+~(Z, V to W~) = 0. It follows that [q] = 0. [] 6.2. The Local structure of support sets Recall that X denotes a rank r Euclidean building. Let Y be a subset of a topo- logical space Z. If [c] e Hk(Z , Y), then we define Support(Z, Y, [c]) s Z -- Y to be the set of points z ~Z- Y such that the image of [c] in the local homology group Hk(Z , Z -- { z }) is nonzero. Support(Z, Y, [c]) is a closed subset in Z -- Y, and contained in the image of the chain c. Lemma 6. ~.. 1. -- Let B be a topologically embedded closed r-ball in X, Y a subset containing OB, and denote by ~ the image of a generator ofH,(B, OB) induced by the inclusion (B, 0B) -+ (X, Y). Then Support(X, Y, ~) = B -- Y. Proof. -- We may apply lemma 6.1.2 since B is a closed (absolute) neighborhood retract. Therefore Support(X, Y, ~t) coincides with Support(B, B n Y, [B]) = B -- Y where [B] denotes the generator of H,(B, OB) which is mapped to ~. [] Now let U be an open subset of X and consider [c] ~ H,(X, U). After subdividing the chain e if necessary, we may assume that the convex hull of each simplex of 0c is contained in U, so that [Str(c)] -= [c]. By 6.1, q ---- Str(c) is carried by a finite union of apartments ~, so [c] is the image of [ca] E H,(~, ~ n U) under the inclusion H,(~, ~ n U) --> H,(X, U). Applying lemma 6.1.2 to the neighborhood retract ~, we find that the inclusion Support(~, ~ n U, [q]) in X coincides with Support(X, U, [c]). Hence we have reduced the problem of understanding Support(X, U, [c]) to a problem about supports in the finite polyhedron ~. Recall that N~ X has a thick spherical building structure with anisotropy poly- hedron A~oa(p) (see section 4.2.2). Lemma 6.2.2. -- Pick p ~U. When ~ > 0 is sufficiently small, log~ maps Support(9 ~, 9 ~ n U, [cl]) n B~(~) isometrically to (u~ C(C~)) n B(~) C C(Z~ X) = C~ X, where the Ci C Z~ X are A~oa(p) chambers and C(C~) C C~ X is the cone over 0 i. Proof. -- The set ~ is a finite union of apartments, so by corollary 4.4.3 when > 0 is sufficiently small log~ maps # n B~(~) isometrically to (u, C~ A~) n B(,) C C~ X, where the A,C ~ are the apartments passing through p. We may assume that U C X\B~(~). Then [q] determines a class [cz] e H,(~ n B~(,), ~ n 0B~(,)). The union w~ Z~ A,C X~ X has a polyhedral structure induced by the thick building atlas dth(p), and this induces a polyhedral structure on the pair (~ n B~(,), ~ n 0B,(a)). 174 BRUCE KLEINER AND BERNHARD LEEB The r-dimensional faces of this polyhedron are (truncated) cones over A~oa(p) chambers in the A~od(P) subcomplex u, X~ A~. C X~ X. Hence the lemma follows from elementary homology theory. [] Corollary 6. B. 3. -- If B is a topologically embedded r-baU in X, then for every p E X\0B there are finitely many A~od(P) chambers C~ C Z~ X so that log~ maps B n Bv(~) isometrically to (u s C(C~)) n B(~) C C~ X for sufficiently small ~ > O. Proof. ~ Let ~ e H,(B, 0B) be the relative fundamental class. Lemma 6.2.1 implies that Support(X, 0B, [~t]) = B\OB and the corollary follows from lemma 6.2.2. [] 6.3. The topological characterization of the llnk If Z is a topological space and z E Z, then we say that two subsets $1, S~ C Z have the same germ at z if $1 n N = $2 r~ N for some neighborhood N of z. The equivalence classes of subsets with the same germ at z will be denoted Germz(Z ). Pick a point x in the rank r Euclidean building X. Consider the collection 5r ) of germs of topological embeddings of R' passing through x ~ X. Let 5r ) be the lattice of germs generated by 5~1(x ) under finite intersection and union. Lemma 6.3.1. -- The lattice 5~,(x) is naturally isomorphic to the lattice ,Yg'Y,, X generated by the A~od(x ) faces of Z, X under finite intersection and union. Proof. -- By lemma 6.2.2 we know that elements of ~9~ correspond to finite unions of ~oa(x) chambers in Z, X. Intersections of A~oa(x ) chambers yield A~0a(x ) faces of Y~, X, so we have a well-defined map of lattices E : 5P~(x) -+~,~E, X by taking each element of 5~,(x) to its space of directions at x (which is a finite union of A~oa(x ) faces). E is injective by Corollary 4.4.3. The image of = contains the apartments in ~Z, X, and since (Z, X, a~ 'th) is a thick spherical building every A~od(X ) face of Y,, X is an intersection of apartments, and hence E is onto. [] 6.4. Rigidity of homeomorphisms In this section we prove the following results about homeomorphisms of Euclidean buildings: Proposition 6.4.1. -- A homeomorphism of Euclidean buildings carries apartments to apartments. Note that homeomorphic Euclidean buildings must have the same rank since the rank is the highest dimension where local homology groups do not vanish. Theorem 6.4.2. ~ Let X, X' be thick Euclidean buildings with topologically transitive affine Weyl group and ? : Y = X � E" ~ Y' = X' � E" a homeomorphism. Then n = n', RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 175 and ~ carries fibers of the projection Y -+ X to fibers of the projection Y' --> X' inducing a homeo- morphism ~ : X ~ X'. Theorem 6.4.3. -- Let X = H~_I _ X~, X' = II~ = 1 X~ be thick Euclidean buildings with topologically transit#e affine Weyl groups, and irreducible factors X~, X~. Then a homeo- morphism ? : X ---> X' preserves the product structure. Theorem 6.4.4. -- Let X, X' be irreducible thick Euclidean buildings with topologically transitive affine Weyl group, and suppose rank(X)/> 2. Then any homeomorphism X -+ X' is a homothety. 6.4. I. The induced action on links Let X, X' be Euclidean buildings, and let ~:X ~ X' be a homeomorphism. Pick a point x in X, and set x' = ~(x) e X'. The homeomorphism ~ induces an iso- morphism of lattices 5P2(x ) ~ 5#2(x') (see section 6.3) and therefore a dimension- preserving isomorphism :r : 3FY~, X -+ 9FY,,, X' of lattices. By proposition 3.8.1 the lattice isomorphism 3Fq~, is induced by an isometry Y~, ? : Z, X --> Z,, X'. 6.4.2. Preservatlon of fiats Consider a singular k-fiat F. Its germ at a point x c F is a subcomplex of 3CZ. X. The image of this subcomplex L under .)FO. is the subcomplex L' associated to the germ of q~(F) in ~r X; L determines a standard (k- 1)-sphere in Z, X. Since 9 :~?, is induced by an isometry Z~ r : Z z X ~ Z,I,I X, L' determines a standard (k -- 1)-sphere in Z~I~I X. This sphere is the space of directions of a singular k-flat F'. q~(F) and F' coincide locally, because their germs coincide. Hence o(F) is a complete simply-connected metric space which is locally isometric to Euclidean k-space E k. Therefore, ~(F) is isometric to E k. 6.4.3, Homeomorph/sms preserve the product structure Let X, X' be Euclidean buildings which decompose as products k I x= 1-Ix,, x'= IIX~ of thick irreducible Euclidean buildings X~., X'~ with almost transitive affine Weyl group. We have a corresponding decomposition of the spherical buildings 1~. X and Y,., X' into joins of irreducible spherical buildings: Y., X = o Y~ X~, Z,, X' = o Y~ X~. We recall that this metric join decomposition is unique, cf. proposition 3.3.3, and therefore for each x e X the isometry Y~. ~ : Y~ X -+ Y~.~ X' decomposes as a join 176 BRUCE KLEINER AND BERNHARD LEEB Z. 9 = o E, ~0~ of isometries E, ~0~ : E,i X~ -+ ~,c,i)oo ) X'oc~l where ~ is a permutation of { 1, . .., k }. In particular, X and X' have the same number of irreducible factors. We claim that the permutation ~ is independent of the point x. To see this, note that any two points y, z ~ X lie in an apartment A and consider the map between apart- ments 91A : A ~ ~(A) (compare section 6.4.2). A parallel family of singular flats in A is carried by ~[A to a continuous family of singular fiats in q~(A); since there are only finitely many parallel families of singular subspaces, we conclude by continuity that q~]A carries parallel singular fiats to parallel singular flats. Consequently the permu- tation ~ is independent of x as claimed. We assume without loss of generality that ~ is the identity. Our discussion implies that a singular fiat contained in a fiber of the pro- jection p~ : X -+ X~ is carried by ~ to a flat in a fiber of the projection p~ : X' -+ X~. Therefore each fiber of the projection p,:X-+ X~ is carried by 9 to a fiber of the projection p':X'-+ X~.'. Hence for each i there is a homeomorphism qh:X~-+ X" such that qh o p~ = p~ o % and it follows that 9 = II, qh. 6.4.4. Homeomorphisms are homothetles in the irreducible higher rank case Let X, X' be as in theorem 6.4.4. Let A be an apartment in X and consider the foliations of A by parallel singular hyperplanes. Since X is irreducible of rank r, we can pick out r + 1 of these foliations ~0, 9 9 -, 9ff, such that the corresponding collection of roots is r-independent (i.e. every subset of r elements is linearly independent) (compare section 3.1). In fact, this property of the root system is equivalent to irreducibility. The image of A under q0 is an apartment A' and the foliations ~ are carried to foliations ~' of A' by parallel singular hyperplanes. Note that these are also r-inde- pendent, since any r-fold intersection of mutually non-parallel hyperplanes belonging to these foliations is a point. Choose affine coordinates xx, ..., x, for A such that the leaves of ~0 are level sets of x x -q- ... + x, and the leaves of the foliation ~ for i >/ 1 t r are level sets of x~. Choose similar coordinates Xl, ... , x, on the target A' so that q~({ x~ = 0 }) = { x~ = 0 } and q~({ Y,x~ = 1 }) = { Y,x~ = 1 }. Consider those leaves in A which contain lattice points. Since q~ maps leaves to leaves one sees by taking successive intersections of these leaves that 9 carries lattice points to lattice points by a homo- morphism. By the same reason 9 induces a homomorphism on rational points and hence, by continuity, an R-linear isomorphism. We now know that ~[a : A ~ A' is an affine map preserving singular subspaces. Angles between singular subspaces are preserved, because the isomorphisms of simplicial complexes ~% are induced by isometries. Hence the simplices { x~ 1> 0, ]~x~ ~< 1 } and { x~ >f 0, ~x~ ~< 1 } are homothetic and ~ is a homothety on A. By considering inter- sections of apartments one sees that the homothety factors are the same for all apart- ments. We conclude that ~ is a homothety. RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 177 6.4.5. The case of Euclidean de Rham factors We now consider Hadamard spaces X ----- Y � E" where Y is a thick Euclidean building of rank r -- n with almost transitive affine Weyl group. Clearly lemma 4.6.7 continues to hold for X, and so do lemma 4.6.8 and the homological statements in section 6.1. Applying the reasoning from section 6.2 we conclude: Lemma 0.4.5. -- Every topologically embedded r-baU in X is locally a finite union IJ, Ci � E" where the C~ C Y are Weyl chambers. It follows that every closed subset of X which is homeomorphic to E" is a union of de Rham fibers, since its intersection with each fiber ofp : X --> Y is open and closed in this fiber. If x ~ X, we may characterize the fiber of p : X -+ Y passing through x as the intersection of all closed subsets homeomorphic to E" which contain x. Now let X' ---- Y' � E"', where Y' is a thick building of rank r' -- n'. If ? : X --> X' is a homeomorphism, then we have r = r' by comparing local homology groups. Since the fibers of the projection maps p : X --> Y, p' : X' -+ Y' are characterized topologically as above, we conclude that ? maps fibers of p homeomorphically onto fibers of p'; therefore n = n' and ~ induces a homeomorphism ~ : Y -+ Y' of quotient spaces. 7. QUASIFLATS IN SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS In this section, X will be a Hadamard space which is a finite product of symmetric spaces and Euclidean buildings. We have a unique decomposition (40) X = E" � II X, where n ~ N O and the Xi are non-flat irreducible symmetric spaces or Euclidean buildings. The maximal Euclidean factor E ~ is called the Euclidean de Rham factor. An apartment is by definition a maximal flat and splits as a product of apartments in the factors. All apartments in X have equal dimension and it is called the rank of X. Singular fiats are defined as products of singular flats in the factors. If the building factors are thick, then singular flats can be characterized as finite intersections of apartments. Note that the only singular flat in E" is E" itself and hence every singular flat in X is a union of de R.ham fibers. 7.1. Asymptotic apartments are close to apartments Proposition 7.1.1. -- Let .~ be a family of subsets in X with the property that for any sequence of sets Q. ~ .~, base points q. ~ Q,, and scale factors d, with ~-lim d, ----- 0% the ultratimit ~-Iim.(d~-l-Q,~, q,) # an apartment in the asymptotic cone o~-Iim,(d~l-X, q,). Then there is a positive constant D so that any set Q ~ .~ is a D Hausdorff approximation of a maximal flat F(Q) in X. 23 178 BRUCE KLEINER AND BERNHARD LEEB Proof. -- Let us consider a single set Q, in .~ and choose a base point q ~ Q. The ultralimit o-lim(n-~.Q,, q) is an apartment in the asymptotic cone ~-lim(n-X.X, q) which contains the base point 9 := (q). Step 1. -- We first show that Q is, in a sense to be made precise, quasi-convex in regular directions. Let x,~yo be a regular segment in m-lim(n-x. O, q) which contains 9 as interior point, xoy~, is the ultralimit of a sequence of segments x,y. in X with endpoints x,,y,~ ~ Q. There is a compact set A C Int(~oa ) which contains the directions of o-all segments xj.. Let F, be a maximal flat containing the segment x,y,. (F, is unique for o-all n.) Pick ~ > 0 so that d(A, 0 A~od) > ,. Denote by D, the diamond- shaped subset of all points p ~ F, so that /.,(p,y,) <~ ~ and /v,(p, x,) ~< ,. Subtemma 7.1.9~. -- There exists r > 0 so that for ~)-all n the sets D, are contained in the tubular r-neighborhood of Q,. Proof. -- We prove this by contradiction: Choose a point z. ~ D, at maximal distance d, from Q, and assume that o-limd, = ~. Then the asymptotic cone {o-lim(d~-a.X, z.) = Cone(X) contains the apartments F' := o-lim, d~-1-F, and F" := o-lim, d~-~.Q. The point z,~ = (z,) is contained in F' but not in F" and therefore F' and F" are distinct apartments in Cone(X). Let z~ x~, (respectively z,j'~) be the ultralimits of the sequences of segments z, x,~ (respectively z--~-~.). By the choice of the points z,, the points x~, and y" are contained in F" u 0o~ F". Since we can extend incoming geodesic segments in apartments according to 4.6.7, we may assume without loss of generality that x'~,y'~ ~ 0o~ F". Let Wx and W~ be the Weyl chambers in Cone(X) centered at z,~ which are spanned by the rays r 1 := z,~ x~, and r 2 := z,~y~. By the choice of r and the definition of D,, the rays r 1 and r~ yield in the space of directions Z,~ Cone(X) interior points of antipodal chambers. Consequently, the union W 1 w W 2 contains a regular geodesic c passing through z~. Since O~ W~ n 0o~ F" contains the regular point r~(~), the chamber 0~o W~ is entirely contained in O~o F". Thus the ideal endpoints c(+ ~) of c are contained in 0| F" and we conclude by 4.6.4 that c C F" and hence z~ ~ F", a contradiction. [] Step 2. -- Suppose q~ ~ Q. and o~-lim n-l.d(q, q~) = O. Sublemma 7.1.8. m We have o-lim d(q,, D,) < oo. Proof. -- The constant sequence q and the sequence q, yield the same point in the ultralimit o-lim (n-a.X, q), which is an interior point of o-lim (n-i.D,, q). Therefore ,. d(q.,D.) (41) o~-um .... 0. d(q., FAD.) RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 179 If m-lim d(q~, D,~) = oo, then :---- o~-lim (d(q., D.)-X-D,,, q.) _~ ~-lim (d(q., D,,)-I.F., q,,) is a complete apartment in o-lim (d(q., D.)-I.X, q.) (by (41)) which lies at unit distance from co-lim q. e (d(q., D.)-X.Q, q.). But the latter is also an apartment in o-lim (d(q., D.) -1. X, q.) and we obtain a contradiction to Corollary 4.6.4. [] We now know that there is an r~ > 0 such that for every R > 0, Q n B~(R) C N.,(D.) for o-all n, for otherwise we could produce a sequence contradicting sublemma 7.1.3. Step 3. -- By steps 1 and 2, we know that there is an r, such that for every R, Q n B~(R) and D. n B~(R) are r,-Hausdorff close to one another for o~-all n. Sublemma 7.1.4. -- For every R > 0, D. n Bq(R) form an ~o-Cauchy sequence (1) with respect to the Hausdorff metric. Proof. -- Suppose X is a symmetric space. Since for c0-all n the sets D. n Bq(R) have mutual Hausdorff distance ~< 2r2, if the sublemma were false we could find Haus- dorff convergent subsequences of { D. } with distinct limits. The limits would be distinct maximal flats lying at finite Hausdorff distance from one another, which is a contradiction. If X is a Euclidean building, then failure of the sublemma would give sequences k., l~ -+ o0 and a radius R so that the Hausdorff distance between Dk. n B~(R) and Dz. n B~(R) remains bounded away from zero. Then the o~-lim(D~, q) and o~-lim(D~,, q) are distinct apartments in the Euclidean building m-lim(X, q) lying at finite Hausdorff distance from one another, contradicting corollary 4.6.4. [] By the sublemma, co-lim D. n B~(R) exists for all R (as an o~-limit of a sequence in the metric space of subsets of B~(R) endowed with the Hausdorff metric) and so we obtain a maximal flat F C X with Hausdorff distance < r 2 from Q. Step 4. -- We saw that each set Qin -~ is the Hausdorff approximation of a maximal flat F(Q.). Denote by d(Q) the Hausdorff distance of Q. and F(Q.). Assume that there is a sequence of sets Q. E .~ with lim d(Q.) = oo. Choose base points u. ~ X so that u. is contained in one of the sets Q.~ or F(Q.) but not in the tubular d(Q.)/2-neighbourhood of the other. Then the apartments ~-Iim d(Q.)-I-Q, and co-lim d(Q.)-I.F(Q..) have finite non-zero Hausdorff distance in the asymptotic cone o-Iim(d(Q.) -a. X, u.). This contradicts 4.6.4. The proof of the proposition is now complete. [5 Corollary 7.1.5. -- There is a positive constant D O = Do(L, C, X, X') such that for any (L, O)-quasi-isometry ~ : X -> X" and any apartment A in X, the image 9(A) is a Do-Haus- dorff approximation of an apartment A' in X'. (1) A sequence x n in a metric space X is r if a subsequence with full r.o-measure is Cauchy. If X is complete, then we define ~o-lim x n to be the limit of this subsequence. 180 BRUCE KLEINER AND BERNHARD LEEB Proof. -- According to proposition 6.4.1, for any sequence of basepoints and any sequence of scale factors ;~, the asymptotic cone qb,o of 9 carries apartments to apart- ments. We can apply proposition 7.1.1 to the collection .~ of all images ~0(A) _~ X' of apartments A in X. [] 7.2. The structure of quasi-flats In this section X will be a symmetric space or a locally compact Euclidean building of rank r, with model polyhedron Amod, and Y will be an arbitrary Euclidean building with model polyhedron A~o d. The goals of the section are the following two results. Theorem 7.2.1. -- For each (L, C) there is a positive real number p such that every (L, C) r-quasiflat 0 in X is contained in a p-tubular neighbourhood of a finite union of maximal flats, Q c Np([JF~ ~ F) where card(~') < p. Corollary 7.2.2. -- The limit set of an (L, C) r-quasiflat 0 in X consists of finitely many Weyl chambers in OTi ~ X; the number of chambers can be bounded by L and C. Lemma 7.2.3. -- Let P C Y be a closed subset homeomorphic to R'. Thus P is locally conical (by corollary 6.2.3), so it has a well-defined space of directions Z~ P for every p e P. We have: 1. If p ~ P then every v ~ Y~ Y has an antipode in E p. 2. If weE p, then there is a ray p-~ c P, ~ e O~l ~ Y such that ~ -~ w. Proof. -- Since P is locally a cone over a E~ P, we have H,_I(Z ~ P) ~ Z, and the inclusion Y~ P ~ Z~ Y induces a monomorphism H,_I(Y,~ P) -+ H,_I(Z ~ Y) since Y~ Y is an r- 1-dimensional spherical building. Now if the first claim were not true, then Z~ PC Y,~ Y would lie inside the contractible open ball B,(~)C Y~ Y, making H,_I(F~ , P) ~ H,_~(Y~, Y) trivial. The second claim now follows from the first by a continuity argument: w is the direction of a geodesic segment contained in P since P is locally conical, and a maximal extension of this segment must be a ray. [] Although we will not need the following corollary, we include it because its proof is similar in spirit to--but more transparent than--the proof of Theorem 7.2.1. Corollary 7.2.4. -- If P C Y is bilipschitz to E" then P is contained in a finite number of apartments. The number of apartments is bounded by the bilipschitz constant of P. Proof. -- Let , ~ A~o d be the barycenter of Amo d, and consider the collection of rays with Amod-direction , contained in P. Since P is bilipschitz to E', a packing argument bounds the number of equivalence classes of such rays (we know that the Tits distance between distinct classes of rays is bounded away from zero (cf. 4.1.2)). RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 181 Let 5~ C 0~l, Y be the (finite) set of Weyl chambers determined by this set of rays, and let S'be the finite collection of flats in Y which are determined by pairs of antipodal Weyl chambers in 5~. We claim that P is contained in Uv ~ ~-F. To see this, note that ifp e P then by lemma 7.2.3 we can find a geodesic contained in P with Amoa-direction 0~ which starts at p. This geodesic has ideal boundary points in 5r so by 4.6.3 the geodesic lies in U~ ~ ~- F. [] Another consequence of lemma 7.2.3 is Corollary 7. ~.. 5. -- Pick ~ ~ A~o a and L, C, ~ > 0. Then there is a positive real number D such that if Q C X is an (L, C) r-quasiflat, y ~ Q, and R > D, then there is z ~ Q with < l a(y, z) - R I< Proof. -- If not, then there is a sequence Q~ of quasiflats, y~ e Qk, and R~ --~ oo such that for very with I a(Y , z,) -- < we hav,/ *. Taking the ultralimit of R~- a. Qk C R~- 1. X we get y~ E Qo C Xo and for every z~ e Qo with[ d(y,~, z,~) -- 1 I < * we have/_(0(y,~ zoo), e) t> ,. But this contradicts lemma 7.2.3 since O~ is bilipschitz to E': we can pick v e Xv, ~ Qo with 0(v) = ~ and find a geodesic segment y~, z,~ C Qo with Y,o zo = v, and for m- all k z~ satisfies the conditions of the lemma. [] Lemma 7.2.3 implies that quasi-flats " spread out ": a pair of pointsy0, z 0 lying in a quasi-flat O C X can be extended to an almost collinear quadruple Yl, Y0, z0, zl while maintaining the regularity of Amoa-directions. To deduce this we first prove a precise statement for Euclidean buildings. Lemma 7.2.6. -- Let ~1 ~ Amoa be a regular point, and let ~1 ~ O. Then there is a 81 E (0, ~1) with the following property. If P C Y is a closed subset homeomorphic to R" and .)'o, zo ~ P satisfy A(0(y--o-~0), ~1) ~< ~a, then there are points yl, z 1 ~ P so that (42) d(zo, z~) = d(yo,y~) = d(yo, Zo) (43) ~-u0(Yl, z0), 2,0(Y0, Zl) > n -- s~ (44) /--(0(y'-~-~x), ~) < ~. The proof requires: Sublemma 7.2.7. -- Suppose x,y, z ~ Y and/~(y, z) = max(D(0(~yy), 0(x--~)) (cf. 3.1). Then x,y, z are the vert#es of a flat (convex) triangle and ~ ~ ~ Y lies on the segment joining~x to a point v ~ ~ Y, where O(v) = 0(-~) and v and.~ lie in a single chamber. Proof of Sublemma 7.2.7. -- Extend the geodesic segments xy, xz to geodesic rays x~l and x~z, ~i ~ OTis X. By hypothesis /,(y, z) ----- max(D(0(~), 0(~-~))) = max(D(0(~l) , 0(~2))) = s ~2). 182 BRUCE KLEINER AND BERNHARD LEEB So x~l ~, determine a flat convex sector S. Note thaty~ andy~, lie in a single chamber of Y~, X since L~(x, ~2) = ~ -- L~(~I, ~2) = n -- max D(0(~I), 0(~.)) = min D(Ant(0(~l)), 0(~2)) = min D(0(y-~), 0(~2)). Hence Axyz bounds a flat convex triangle T C S, and so y~ lies on the geodesic segment which has endpoints .~ and y~,. [] Proof of Lemma 7.2.6. -- Pick z I~P so that z oz 1CP, d(zo, zl) =d(yo, Zo) , 0(z--~l) = e, and z o z 1 ~ Y,,o Y lies in a chamber antipodal to ZoYo; similarly choose Yl e P so that YoYl C P, d(yo,yl) =- d(yo, Zo), 0(y--o~l) = Ant(e), and ~ ~ Y'~o Y lies in a chamber antipodal toy o z o. Applying sublemma 7.2.7 we conclude that Zo, Yo, zl are the vertices of a flat convex triangle, and Yo zl e E~o Y lies on the segment joining Yo zo to v ~ E~o Y where 0(v) = 0(z-~-~) = e and v and ~ lie in the same chamber. In particulary o zl andyoyl lie in antipodal chambers of Z~o Y, so applying lemma 7.2.7 again, we find that 0(y-]-~) lies on the segment joining 0(y--~-~x ) to 0(y-~-~) = e. Yl and zx clearly satisfy the stated conditions since 7__~o(yl, Zo) /> L~o(yl, Zo) = n -- L(0(y--~o), e) i> ~ -- 81 > ~ -- ~1 and ~-,o(Y0, zl) 1> /,o(Yo, zl) = n --/(0(y-~-~o), e) >/ n -- 81 > n -- ~1. [] Corollary 7.2.8. -- Let e2 ~Amoa be a regular point, and let L, C, ~2 > 0 be given. Then there are D 2 > O, 82 ~ (0, ~2) with the following property. If Q c_ x is an (L, C) r-quasiflat, and Yo, Zo ~ Q satisfy (45) d(yo, zo) > D2, /--(0iy-~-~o), e2) ~< 82 then there are points Yl, Zl ~ Q so that (46) I d(zo, - d(yo, zo) I, I d(yo,Y~) -- d(yo, Zo) I< ~2 d(yo, zo) (47) Z~o(yl, Zo) , Z,o(y0, zl) > r~ -- ~, (48) /_(0(y-~--~), e2) < 8,. Proof. -- Let 8~, X2 be the constants produced by the previous lemma with el = e2, ~1 = ~" We claim that when Yo, Zo e Q and /(0(y-o-~o), e~) < ~. and d(yo, Zo) is sufficiently large, then there will exist points Yl, zl satisfying (46), (47), (48). But this follows immediately from the previous lemma by taking ultralimits. [] By applying corollary 7.2.8 inductively we get Corollary 7.2.9. -- With notation as in corollary 7.2.8, there are sequences Yi, z~ ~ Q, i >1 1 such that the inequalities (45), (46), (47), (48) hold when we increment all the inc~ces on the y's and z's by i. ______> RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 183 Lemma 7.2.10. -- Fix ~. > O, and consider all configurations (y, z, F) where y, z ~ X, /__(0()-~), 0 A~od) t> ~, and F C X is a maximal flat. Then there is a D 3 such that the fraction of the segment )-~ l~ng outside the tubular neighbourhood ND~(F ) tends to zero with v (y, z, F) d~ max(d(y, F)/d(y, z), d(z, F)/d(y, z)). Proof. -- Recall that the distance function d(F, . ) is convex, so if the lemma were false there would be sequences y,, zk, w ~ e X, FkC X, with /_(O(y--~-~k),OAmoa)>i ~x, d(y, z) ~ oo, w, s y, z, with d(w,,y,), d(w,, z,) > ~ d(y,, z,), v~(y,, z,, F,) ~ 0 but d(w,, F,) ~ oo. Let p,, q,, r, e F, be the points nearest y,, w,, z, respectively. By various triangle inequalities and property (39) from section 5.2 we have 7/,~(p~,y~), ~f~(r~, z~) ~ 0 and /(0(p--~--~), 0(y--~-~)), /_(0(q-~-~), 0(y--~-~))~ 0. There- fore if we set R~ = d(q~, w~) and take the ultralimit of (R~-~.X, q~) we will get a confi- guration q,,, w,o ~ X,o , an apartment F,o C Xo, , and ~, ~ e 0~t, X,~ so that q,, is the point in F,o nearest to w,o, (q,o ~, q,,) ---- o~-lim(q--~p-~, q~), (q., ~, q,o) = c0-1im(q-~, q~), (w,o ~, q,.) ----- oMim(w~, q~), (w~ ~, q,,) = c0-1im(w~ z~, q~). In particular, the rays w~ ~ and w~ ~ fit together to form the geodesic oMimy~ z~ and /(0(~), 0 Amoa) >/ ~. But this contradicts corollary 4.6.4. [] Corollary 7.2.1t. -- Fix o~ 3 ~ A~o a. Then there are constants ~, v4, D4 such that if 1. y,, z, ~ X, i >>. 0 are sequences which satisfy (45), (46), (47), (48) (when subscripts are incremented by i) with % < ~4, d(yo, Zo) > D4. 2. A maximal flat F C X satisfies d(yk, F), d(zk, F) < v4 d(y~, z~) for some k. Then d(y,, F), d(z,, F) < v 4 d(y,, z,) for all 0 ~ i <~ k. Proof. ~ If v~ is sufficiently small, then the trisection points ~, ~" of any sufficiently longsegment~z C Xwith/_(0()-~), 0 Amoa) >/ ~, max(d(y, F) ld(y , z), d(z, F)/d(y, z) ) < v 4 will satisfy max(d(y,F)/d(y,'~),d('~, F)/d(~,~'))~ v 4 by lemma 7.2.10. If we take ~4 ~ v4 then /(0(yT-~,), 0 Amoa) will be bounded away from zero and y,_~, z,_~ will lie close to the trisection points of y, z, so corollary 7.2.11 follows by induction on k--i. rq Proof of Theorem 7.2.1. Step 1. -- Fix ~4 e Amoa, and let ~s, vs, D6 be the constants produced by corol- lary 7.2.11 with a3 ~ a,. Let D6, 86 be the constants given by corollary 7.2.8 with = a4, ~----%. Finally, let D, be the constant produced by corollary 7.2.5 with = a4, g = min(86, 1/2). Setting D s ----- max(I)5, D6, D~), for eachy 0 ~ Q. we may find a z 0 E Q with D, < d(yo, zo) < 2 D 8 so that /__(0(y0, z0), a4) < 86 (by corollary 7.2.5). By corollary 7.2.9 we may extend the pair Yo, z0 E Q. to a pair of sequences y,, z, satisfying (45)-(48) with a, = a4, r = cs- Then any maximal fiat F C X with 184 BRUCE KLEINER AND BERNHARD LEEB d(y~, F), d(z~, F) < '~s d(y~, zk) for some 0 ~< k < oo satisfies d(y~, F), d(z~, F) < v b d(y~, z~) for all 0 ~< i <~ k by corollary 7.2.11; in particular (49) d(yo, F) < v s d(yo, zo) < 2~ D,. We may assume in addition that a s is small enough so that (50) 2 d(y~_l, z~_l) < d(y~, z~) < 4 d(y~_~, z~_~), and (51) d(y~,y~__~), d(z~, z~_a) < 2 d(y~_a, z~_~). It follows that (52) max(d(y~,yo), d(zi,yo) ) < 2 d(y~, z~) for all i. Step 2. -- Fix q ~ Q and set v 6 ---= vJ16. For each R pick a covering of Bq(R) n Q, by v6 R-balls { B~i(v 6 R)} with minimal cardinality; the cardinality of this covering can be bounded by r and the quasiflat constants (L, (3). For each pair p~, pj of centers pick a maximal flat containing them, and denote the resulting collection of maximal flats by o~r. Claim. -- If y o e Q, then d(yo, [JF~s,'~ F) < 2v 5 D s for sufficiently large R. Proof of claim. -- We will use the sequences y~, z~ constructed in step 1 and esti- mate (49). Take the maximal i such that y~, z~ ~ B~(R). Then max(d(y,+a, q), d(z,+~, q)) > R =~ max(d(y,+~,y.), d(z,+a,yo) ) > R -- d(q, yo) * d(y,+~, Z,+l) >/2(R -- d(q, yo)) by (52) a(y,, z,) >/I(R - d(q, yo)) by (50). Since #'a~ contains a maximal flat F with 8v, R ~ 1 (R -- d(q, yo)) d(y,, F), d(z,, F) < v e R = R -- d(q, yo)]'-8 <~ 8V6 (R _ ~q, yo~) d(Y,, Z,) < 7 R - yo/ e(Y" RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 185 Therefore for sufficiently large R there is an F ~'~ and k such that d(yk, F), d(zk, F) < ~5 d(Yk, zk), so d(yo, F)< 2v 5 D s as claimed. [] Proof of Theorem 7.2.1 concluded. -- We may now take a convergent sub- sequence of the ~'g's, and the limit collection o~" satisfies Q c N2,s([JF~F ) and card(~) ~< lira sup card(~-r) which is bounded by r and (L, G). [] Proof of Corollary 7.2.2. -- By theorem 7.2.1 there is a finite collection ~- of maximal flats so that O lies in a finite tubular neighbourhood of [.J, E ~ F. The limit set of each F ~ ~- is its Tits boundary 0Tt ~ F, which is an apartment of 0~ X. The union of these apartments gives us a finite subcomplex ff C 0~ X which is a union of closed Weyl chambers. Clearly LimSet(Q) _ if; we will show that if ~ ~ LimSet(Q,) then ~ lies in a closed Weyl chamber C C LimSet(Q). We have qk ~ Q. such that *qk ~*~ in the pointed Hausdorff topology. Consider [JF~ F. Any ultralimit c~ -1" ([-]F~ F), ,) is canonically isometric to the Euclidean cone over ft. The set eo-lim(R~-~-Q, *) embeds in ~-lim(R~ -1. ([J~ F), ,) as a bilipschitz copy of E'; by the discussion in section 6.2, ~-lim(R~-X.Q, ,) is the cone over a collection of closed Weyl chambers in ft. In par- ticular co-lim ,q, -~ *,o q~ lies in a closed Weyl chamber contained in ~0-1im(R~ -~ . O, ,), so the corresponding Weyl chamber of ff is contained in LimSet(Q,), and it contains ~. [] 8. QUASI-ISOMETRIES OF SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS In this section our goal is to prove theorems 1.1.2 and 1.1.3 stated in the introduction. Let X, X', and 9 be as in theorem 1.1.2. By corollary 7.1.5, 9 carries apartments close to apartments; in particular, X and X' have the same rank r. 8.1. Singular flats go close to singular flats Lemma 8.1.1. -- For any R :> 0 there is an D(R) ~ 0 such that if F is a singular flat in X and d(F) is the collection of apartments containing F, then ['IAe~r ) C ND~(F ). Proof. -- It suffices to verify the assertion for irreducible non-flat spaces X. Consider first the case where X is a symmetric space. The transvectlons along geodesics in F preserve all the flats containing F. Hence, if there is a sequence x. e [']AE~IF)NR(A) with d(x,, F) tending to infinity, then we may assume without loss of generality that the nearest point to x, on F is a given point p. The segments px, 24 186 BRUCE KLEINER AND BERNHARD LEEB subconverge to a ray p~ which lies in ['IAe~F)N~(A) and is orthogonal to F. Since for each apartment A e ad(F), we have p cA and the ray p~ remains in a bounded neighbourhood of A, it follows that p-~ C ['] A ~ aiFI F. Hence rlx e ~,cF~ F contains a (k + 1)- flat, which is a contradiction. Assume now that X is an irreducible thick Euclidean building with cocompact affine Weyl group. Consider a point x e X\F and let p ~ F be the nearest point in F. Then u := p~ e Z~ X satisfies /~(u, ~ F) >/ n]2. We pick a chamber C in X~ X con- taining u and choose a face a of C at maximum distance from u. Denote by v the vertex of C opposite to ~. By our assumption, diam(A~oa) < n/2 and therefore v r Y,~ F. Since F is a finite intersection of apartments, lemma 4.1.2 implies Y,~ F = ['IAE~r ~ X A and there is an apartment A with F C A C X and v r X~ A. 2g A is then disjoint from the open star of v, and so d(u, Z~ A) >~ d(u, ~)/> *t o > 0 where a0 depends only on the geometry of A~o a. If x ~ N~(A) then angle comparison implies that d(x, F) ~< R/sin ~0 and our claim holds with D(R) = R/sin ot 0. This completes the proof of the lemma. Proposition 8.1.2. -- For every apartment A C X, let A' C X' denote the unique apartment atfinite Hausdorff distance from ~(A). There are constants D0(L , C, X, X') and D(L, C, X, X') so that/fF = ~xal, AC X is a singular flat, then 1. r C f'la~. N.o(A'), 2. The Hausdorff distance da(r ['lx~ F N.o(A')) < D, 3. There is a singular flat F'C [']a~, ND0(A') with d,(O(F), V')< D. In particular, two quasi-isometries r ~2 : X --~ X' inducing the same bijection on apartments induce the same map of singular flats up to 2D-Hausdorff approximation. Proof. -- Let F and d(F) be as in the previous lemma. By corollary 7.1.5, for every apartment A G X, ?(A) is Do-Hausdorff close to an apartment in X' which we denote by A'. Thus ~(F)C f]AE~'~FI N-o(A')" Sublemma 8.1.3. -- For each d>~ D o there exists a constant D x = DI(L , C, d) > 0 with the property that ~xedcFI Na(A') lies within Hausdorff distance Dlfrom q~(F). Proof. -- Pick a quasi-inverse ,-x of q~. For each point y ~ f'lAe~Fj Na(A' ) and each A ~ d(F), q~-Xy is uniformly close to ~-1 A'. But q~-I A' is uniformly Hausdorff close to ~-x ~A and therefore to A. Lemma 8.1.1 implies that Y has uniformly bounded distance from F. [] Proof of Proposition 8.1.2 continued. -- Fixing A 0 ~d(F), we conclude that C := (f]Ac~r N~Do(A')) r3 A o is a convex Hausdorff approximation of ~(F). Sublemma 8.1.4. -- Let C C E l be a convex subset which is quasi-isometric to E ~. Then C contains a k-dimensional affine subspace. RIGIDITY OF Q.UASI-ISOM'ETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 187 Proof. -- Fix q c C and let C _ C be the convex cone consisting of all complete rays starting in q and contained in C. For any sequence X,, ~ 0 of scale factors, the ultralimit r q) is isometric to (~. Therefore C is homeomorphic to 1r k and hence isometric to loP. [] Proof of Proposition 8.1.2 continued. -- It follows that 9(F) is uniformly close to a flat F in X'. Since 9,~ carries singular flats to singular flats, 0~l ~ F is a singular sphere in 0Tl ~ X'. X' has cocompact affine Weyl group, so F lies within uniform Hausdorff distance from a singular flat F'. [] 8.2. Rigidity of product decomposition and Euclidean de Rham factors We now prove theorem 1.1.2. The product decompositions of X and X' cor- respond to a decomposition of asymptotic cones (53) = E" x rl = E-' � rI xL where the X~o, X'j~ are irreducible thick Euclidean buildings. They have the property that every point is a vertex and their affine Weyl group contains the full translation subgroup, in particular the translation subgroup is transitive. We are in a position to apply theorems 6.4.2 and 6.4.3: The Euclidean de Rham factors of X and X' have equal dimension, n = n', and X, X' have the same number of irreducible factors. After remumbering the factors if necessary, there are homeomorpkisms (?o)~:Xi,~-~-~-',~ such that (90), o Ao = o where p, : X -+ X~ and p~ : X' -+ X~ are the projections onto factors. Now let F be a singular flat which is contained in a fiber ofp,. By proposition 8.1.2, 9(F) is uniformly Hausdorff close to a flat F' C X'. Since F',o C X',~ is contained in a fiber of p,~, F' must be contained in a fiber of p~. Any two points in a fiber p~l(x,), xi z X,, are contained in some singular flat F C p~-l(x,) and consequently 9 carries fibers of p~ into uniform neighbourhoods of fibers of p~. Since an analogous statement holds for a quasi-inverse of 9, we conclude that 9 carries p~-fibers uniformly Hausdorff close to p~-fibers and so there are quasi-isometries %:X~--> ~ so that 9 o A = P~ o 9 holds up to bounded error. This concludes the proof of Theorem 1.1.2. 8.3. The irreducible case In this section we prove theorem 1.1.3. Note that theorem 1.1.2 implies that X' is also irreducible, with rank(X) = rank(X'). 188 BRUCE KLEINER AND BERNHARD LEEB 8.3.1. Quasi-isometries are appro=imate homotheties We recall from proposition 7.1.5 that 9 carries each apartment A in X uniformly close to a unique apartment in X' which we denote by A'. We prove next that in our irreducible higher-rank situation the restriction of ~b to A can be approximated by a homothety. As a consequence, the quasi-isometry ~ is an almost homothety. This parallels the topological result in section 6.4.4. Proposition 8.3.1. -- There are positive constants a = a(~b) and b = b(L, C, X, X') such that for every apartment A C X exists a homothety tF A : A ~ A' with scale factor a which approximates ~[, up to pointwise error b. 4t Proof. -- If we compose ~IA with the projection X' -+ A', we get a map ~ A : A ~ A' which, according to proposition 8.1.2, carries walls to within bounded distance of walls. Parallel walls in A are carried to Hausdorff approximations of parallel walls in A'. Moreover, due to our assumption of cocompact affine Weyl group, each hyper- plane parallel to a wall is carried to within bounded distance of a wall. By lemma 3.3.2 exist r + 1 singular half-spaces in A which intersect in a bounded affine r-simplex with non-empty interior. Consider the collection cg of hyperplanes in A which are parallel to the boundary wall of one of these half-spaces. Any r pairwise non-parallel hyperplanes in ~ lie in general position, i.e. intersect in one point. Hence we may apply lemma 8.3.3 below to the collection ~' and conclude that tFk is within uniform finite distance of an affine transformation WA : A ~ A'. Since ~ is a homothety on asymptotic cones by the discussion in section 6.4.4, it follows that tF A is a homothety: For suitable positive constants a A and b we therefore have [ d(~x(Xl), ~Fx(x2)) -- ax d(xx, x,) [ <<. b V x~, x 2 ~ a and b depends on L, C, X, X' but not on the apartment A. To see that the constant a x is independent of the apartment A note that for any other apartment A 1 C X there is a geodesic asymptotic to both A and A 1. It follows that a& = a A, [] Corollary 8.3.2. -- There are positive constants a = a(~b) and b = b(L, C, X, X') such that the quasi-isometry ~ : X -+ X' satisfies [ d(q)(xl), @(x2)) -- a.d(xl, x2) I <~ b V xl, x2 e X. Here L- I <~ a <~ L. Proof. -- This follows from the previous proposition, because any two points in X lie in a common apartment. [] Lemma 8.3.3. -- For n >1 2, let ~0, 9 .., ~, e (R")* be a collection of linear funetionals any n of which are linearly independent, and let,~ be the collection of affine hyperplanes { ~- l(c) }c~a- There is a function D(C) with lirnc_~0 D(C) = 0 satisfying the following: If ~ : R" ~ It" RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 189 is a locally bounded map such that for all H ~ ~,~dj, q~(H) C Nc(H' ) for some H' ~ ~, then there is a an affine transformation ~o with scalar linear part which preserves the hyperplane families .~ such that d(% %) < D(C). Proof. -- After applying an affine transformation if necessary we may assume that ~0 = Y~=l x,, % = xj for 1 .<j~< n, and ~0(0) = 0. There is a C2 eR such that the image of each k-fold intersection of hyperplanes from [J, ~ lies within the C 2 neigh- bourhood an intersection of the same type. In particular, for each 1 ,< j ~< n, ~ induces a (C3, *3) quasi-isometry % of the j-th coordinate axis, with %(0) = 0. It suffices to verify that each % lies at uniform distance from a linear map since ~0 lies at uniform distance from II~.= 1%. Also, it suffices to treat the case n = 2 since for each 1 ~< j ~< n we may consider the (C4, ,4)-quasi-isometry that q~ induces on the x~x i coordinate plane, and this satisfies the hypotheses of the lemma (with somewhat different constants). We claim there is a C 5 such that for y, z in the first coordinate axis, we have [ qh(Y + z) -- (q~l(Y) + ~l(Z)) [< C5- To see this first note that when C equals zero the additivity can be deduced from a geometric construction involving 6 lines and 6 of their intersection points. When C ~ 0, the same construction can be performed with uniformly bounded error at each step. By lemma 8.3.4 below, qh and analogously % lies at uniform distance from a linear map. [] Lemma 8.3.4. -- Suppose +:R-->R is a locally bounded function satisfying I+(Y + z) -- +(y) -- +(z) ]<~ D for ally, z sR. Then ]+(x) -- ax ]<~ D for some a eR. Proof. -- Since ] +(2") -- 2+(2 "-1) [~< D, the sequence (+(2")/2") is Cauchy and converges to a real number a. Let x > 0 and choose numbers q, ~ N and r, ~ R with I r, [ ~< x such that 2" = q, x + r,. Then I +(2") -- q. +(x) -- +(r.) I~< (q. + 1) D and hence, using that + is locally bounded, I +(~") ~x +(r,) x (q,+ 1)x D. x -- ~(x) 2" ~< 2" -'r -+1 "-*'0 "-'1 When n tends to infinity, we obtain in the limit lax -- +(x) [<<. D. Similarly, there is a real number a_ such that for all x < 0 we have ] a_ x -- + (x) I ~< D. Since [+(x) + +(-- x) l~< D + I +(0)I, it follows that a = a_. [] Proof of Theorem 1.1.3 concluded. -- By corollary 8.3.2 we may scale the metric on X' by the factor 1]a so that 9 becomes a (1, A/a) quasi-isometry. Applying propo- 190 BRUCE KLEINER AND BERNHARD LEEB sition 2.3.9 we conclude that 9 induces a map O.~ 9 : O.~ X -+ ~ X' which is a homeo- morphism of geometric boundaries preserving the Tits metric. By the main result of 3.7, 0~ 9 gives an isomorphism of spherical buildings 0~ 9 : (0~t~ X, Amos) ~ (0~t,, X', ~o~), after possibly changing to an equivalent spherical building structure on O~t,. X'. Consequently, for every 8 e Amoa, O.~ maps the set 0-a(8)C 0~l~,X to the corres- ponding set 0'-a(8)C Ow~X', and ~[0-~1 is a cone topology homeomorphism. When is a regular point, the subsets 0-a(8) C0~.X and 0'-a(8) C0~t~,X ' are either manifolds of dimension at least 1 or totally disconnected spaces by sublemma 4.6.9, depending on whether X and X' are symmetric spaces or Euclidean buildings. Therefore either X and X' are both symmetric spaces of noncompact type, or they are both irreducible Euclidean buildings with Moufang boundary. In the latter case we are done by theorem 8.3.9; when X and X' are both symmetric spaces we apply propo- sition 8.3.8 to get a homothety ~0 : X -+ X' with O., ~o = O~ ~. By proposition 8.1.2, d(r Co(V))< D for every vertex v e X, and since the affine Weyl group of X is cocompact the vertices are uniform in X, and so we have d(~, r < D'. Hence r is an isometry. [] 8.3.2. Inducing isometrles of/deal boundaries of symmetric spaces We consider a symmetric space X of non-compact type and denote by G the identity component of its isometry group. Sublemma 8.3. ft. -- Let F C X be a maximal flat and let ~F : X ~ F be the nearest point retraction. Given a compact set K C Int(A~od) and e > O, there is a ~ > 0 such that if p E X, x e F, 0(-~) ~ K, and /~(x, ~(p)) > ~/2 -- ~, then d(p, F) < r Proof. -- Note that as q moves fromp to rcF(p) along the segment ~-F(P) P,/_=(x, ~(p)) increases monotonically. If the sublemma were false, we could find a sequence Pk e X, x~ e F so that L~k(Xk,~F(pk))-+~/2 and d(pk, F ) 1> e. Since /,~l~k~(xk,pk)= ~/2, triangle comparison implies that [Pk ~F(Pk) [/1P~ X~ I ~ 0. Hence by taking q~ epk 7rx,(p~) with d(qk, F)----e we have /~(Pk, q~)-+0, so d~od(0(q--~-~k), K)-+0. Modulo the group G, we may extract a convergent subsequence of the configurations (F, q--~-~k) getting a maximal flat F, a point q~ with d(q~, F)= e, and x| e0| F such that /-q=(xoo, ~F(qoo)) ----~/2, and 0(x~) e K. This is absurd. [] Sublemma 8.3.6. -- Let F, be a sequence of maximal flats in X so that 00, F~ ~ 0~, F where F is a maximal flat, i.e. for each open neighborhood U of O~ F in O~ X with respect to the cone topology, 0~o F, is contained in U for sufficiently large i. Then F, ~ F in the pointed Hausdorff topology. Proof. -- Let 4, ~ e 0= F be antipodal regular points and choose points ~-, ~, e 0.~ F~ so that ~ -> ~ and ~ --> B. Then for x e F we have L,(~, ~) --+ r~ and consequently RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 191 L,(nFI x, ~.) ~ ~/2, L,(~Fi x, ~,) ~ ~/2. Applying sublemma 8.3.5, we conclude that d(x, F,) ~ 0. The claim follows since this holds for all x E F. [] Lemma 8.3.7. -- Let 0oo : G -+ Homeo(O~ X) be the homomorphism which takes each isometry to its induced boundary homeomorphism. Then 0oo is a topological embedding when Homeo(0~ X) is given the compact-open topology. Proof. M The homomorphism 0~o is continuous, because the natural action of G on 0~o X is continuous. To see that 0~o is a topological embedding, it suffices to show that if g, e G is a sequence with 0~(g,) ~ e s Homeo(O~o X), then g, -~ e e G. Let x be a point in X and choose finitely many (e.g. two) maximal flats F1, ... , F, with F in... nF k={x}. Since O~(g,)-+eeHomeo(0~oX),0| Fj converges to 0~F~ in the sense that for each open neighborhood Uj of 0~ Fj in 0oo X with respect to the cone topology, 0~o g, Fj is contained in U~ for sufficiently large i. By the previous sublemma we know that g, F~ ~ Fj in the pointed Hausdorff topology. [] Proposition 8.3.8. -- Let X and X' be irreducible symmetric spaces of rank at least 2. Then any cone topology continuous Tits isometry + : 0~ X -~ o~,~ X' is induced by a unique homothety ~:X-+ X'. Proof. -- We denote by G (resp. G') the identity component of the isometry group of X (resp. X'). By lemma 8.3.7 the homomorpkisms 0= : G ~ Homeo(0~ X) and 0~o : G' -+ Homeo(0| X') are topological embeddings, where Homeo(0| X) and Homeo(0~ X') are given the compact-open topology. According to [Mos, p. 123, cor. 16.2], conjugation by + carries 0oo G to 0~o G'. Hence + induces a continuous isomorphism G-+ G'. Such an isomorphism carries (maximal) compact subgroups to (maximal) compact subgroups and it is a classical fact that the induced map ~" : X -~ X' of the symmetric spaces is a homothety. The isometry + and the induced isometry 0Tl~, ~ at infinity are G-equivariant with respect to the actions of G on 0Tl ~ X and 0~m , X' and we conclude that 0~v. ~" = ~b. [] 8.3.3. (1, A) quasi-isometries between Euclidean buildings Here we prove Theorem 8.3.9. -- Let X, X' be thick Euclidean buildings with Moufang Tits boundary, and assume that the canonical product decomposition of X has no 1-dimensional factors (1). Then for every A there is a constant C so that for every (1, A) quasi-isoraetry ~ : X --~ X' there is an isometry ~o : X ~ X' with d(ep, ~o) < C. (x) The statement is false for (1, A) quasi-i, ometries between tree*. 192 BRUCE KLEINER AND BERNHARD LEEB The proof of Theorem 8.3.9 combines corollary 7.1.5 and material from sec- tions 3.12 and 4.10. We first sketch the argument in the case that X and X' are irre- ducible, of rank at least 2, and have cocompact affine Weyl groups. Let (B, Amoa) be a spherical building. Attached to each root (i.e. half-apartment) in B is a root group U~ ~ Aut(B, Amod) (see 3.12). Remarkably, when B is irreducible and has rank at least 2, the U,'s--and consequently the group G ~ Aut(B, Amod) generated by them--act canonically and isometrically on any Euclidean building with Tits boundary B (see 4.10). Now let (B, Amoa) = (0T~t,X, Amoa). If ~:X ~ X' is an (L, A) quasi-isometry, then by 2.3.9 we get an induced isometry 0Tit, 9 : 0~tt, X -+ 0Tit, X', so the group G _ Aut(B, Amoa) acts on 0r ~ X, 0Tit, X', and hence on X and X'. By comparing images of apartments (and using the quasi- isometry q)), one sees that a subgroup K c_ G has bounded orbits in X if and only if it has bounded orbits in X'. Because B is Moufang (3.12) the maximal bounded sub- groups IVIC G pick out " spots" v,, eX and bM eX' (proposition 4.10.6), and the resulting 1-1 correspondence between the spots of X and the spots of X' determines a homothety (I) 0 : X --+ X' with 0Tlt~ (I) 0 = 0~,t~ ~. Proof of Theorem 8.3.9. Step 1. -- Reduction to the irreducible case. Lemma 8.3.10. -- Every (1, A) quasi-isometry ~ : FZ -+ E" lies within uniform distance of a homothety. For every distance function d:E'-+ E' the function d o q~ lies within uniform distance of a distance function. By taking limits we see that for every Busemann function b : E' -+ E', b o q~ is uniformly close to a Busemann function. But tile Busemann functions are affine functions, so c? is uniformly close to an affine map q~0. Obviously q~0 is an isometry. [] By corollary 7.1.5, there is a constant D(A, X, X') such that the image of every apartment A C X is D Hausdorff close to an apartment A'C X'. Composing (I)!A with the projection onto A' we get a map which is uniformly close to an isometry ~IJ'A : A -+ A'. Hence ifF C A is a flat, then ~)(F) C X' is uniformly Hausdorffclose to the flat ~A(F) C A'. Therefore we may repeat the reasoning of 8.2 to see that if X = IIxi, X' = IIX'~ axe the decompositions of X and X' into thick irreducible factors, tllen after reindexing the factors X'j there are (1, A) quasi-isometrics ~t:Xi-+ X~ so that q~ is uniformly close to II~ (A depends only on the quasi-isometry constant of 9 and X, X'). Hence we are reduced to the irreducible case. Step 2. -- The buildings X and X' are irreducible. The affine Weyl groups W~, W~r of X, X' are either finite or cocompact, since their Tits boundaries are irreducible. If Win is finite then it has a fixed point, so all apartments intersect in a point p e X and X is a metric cone over 0T~t~ X. If~ e Am~ is a regular point, then 0-1(~) C 0Tit, X is clearly discrete in the cone topology. On the other hand, if W,~ is cocompact then RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 193 0-a(a) C Ox~ X is nondiscrete since any regular geodesic ray p~c A can branch off at many singular walls. Since 9 induces a homeomorphism of geometric boundaries 0~o ~ : 0~ X -~ 00~ X' by 2.3.9, and this induces an isomorphism of spherical buildings 0~, ~ : 0~t~ X ~ 0~t, X', either X and X' are both metric cones, or they both have cocompact affine Weyl groups. If they are both cones, we may produce an isometry ~0 : X ~ X' by taking the cone over Ox~t~ 9 : 0~t . X ~ 0T~ ~ X'. This induces the same bijection of apartments as ~, and lies at uniform distance from ~ by lemma 8.3.10. Step 3. -- The buildings X and X' are thick, irreducible, and have cocompact affine Weyl group. Letting G C Aut(0~t~ X) ~ Aut(0~t. X') be the group generated by the root groups of 0T~t~ X, we get actions of G on 0T~, X, 0T~, X', and by 3.12.2 actions on X and X' by automorphisms as well. Lemma 8.3.11. -- A subgroup B C G has bounded orbits in X if and only if it has bounded orbits in X'. Proof. -- We show that if K has a bounded orbit K(p) = { gp [ g e K } C X then K has a bounded orbit in X'. Let p E X be a vertex, let ~-~ be the collection of apartments passing through p, and let "~'xc~l = [J0EK~-gp 9 ~'~r~l is a K-invariant collection of apartments in X, and when R> Diam(K(p)) we have p ~ ~Ae~x(p)N~(A). Let ~(~-,) and ~(~'K(~)) denote the corresponding collections of apartments in X'. Then r is K-invariant, and r e [~x,e~ar,(p~, Ng+c,(A'), where C 1 is a constant such that for every apart- ment ACX, the Hausdorff distance da(r By proposition 8.1.2, [']x,e~-p~NR+c~(A') is bounded. Thus f]A,e~s~,(p~NR+c,(A') is a nonempty K-invariant bounded set. [] Proof of Theorem 8.3.9 continued. ~ By proposition 4.10.6 we now have a bijection Spot(~) : Spot(X) -+ Spot(X') between spots in X and X' via their correspondence with maximal bounded subgroups in G. Moreover by item 2 of proposition 4.10.6 for every apartment AC X, we have Spot(~)(Spot(A))= Spot(A') where A'C X' is the unique apartment with 0TI,. A' ---- 0~l~, ~(0~l~, A). Since by item 3 of proposition 4.10.6 Spot(@) ]s~t,x, : Spot(A) -+ Spot(A') is a homeomorphism with respect to half-apartment topologies we see that X is discrete if and only if X' is discrete. Case 1: Both X and X' are non-discrete, i.e. their affme Weyl groups have a dense orbit. In this case Spot(A) = A, Spot(A') = A', and Spot(~)[x : A ~ A' is a homeomorphism 25 194 BRUCE KLEINER AND BERNHARD LEEB since the half-apartment topology is the metric topology. By item 3 of proposition 4.10.6 Spot(O)] x maps singular half-apartments H C A with 0~l~,H = a to singular half- apartments Spot(O) (H) C A' with 0~l~(Spot(O ) (H)) = 0Ti ~ O(a). By considering infinite intersections of singular half-apartments with Tits boundary a C 0~, A, it follows that Spot(O) carries all half-spaces H C A with 0~l~H = a to half-spaces Spot(O) (H) with 0Tlt~(apot(O ) (H))= OTltaO(a ). By considering intersections of half-spaces H with opposite Tits boundaries, we see that Spot(O) carries hyperplanes whose boundary is a wall m C 0~lt~A to hyperplanes in A' with boundary 0T~t~O(m) C 0~t.A'. By section 6.4.4 it follows that O0dZSpot(O):X~X' is a homothety and 0~l~, 9 o = 0~lt, O. Case 2: X and X' are both discrete. In this case A and A' are crystallographic Euclidean Coxeter complexes; Spot(A) and Spot(A') coincide with the 0-skeleta of A and A'. Again by item 3 of proposition 4.10.6, if S C A is either a singular subspace or singular half-apartment, then there is a unique singular subspace or singular half-apartment S'C A' so that Spot(O) (S ca Spot(A)) = S' ca Spot(A'). k + 1 spots So, ..., sk ~ Spot(A) are the vertices of a k-simplex in the simplicial complex if and only if they do not lie in a singular subspace of dimension < k and the intersection of all singular half-apartments containing { so, ..., s~ } contains the k + 1 spots s~. Hence Spot(O) ]spot A : Spot(A) -~ Spot(A') is a simplicial isomorphism and hence is induced by a unique homothety A ~ A'. It follows that Spot(O) :Spot(X) -~ Spot(X') is the restriction of a unique homothety 9 0 : X -~ X' with 9zl~ Oo = 9~lt~ O. Since vertices are uniform in X, we may apply proposition 8.1.2 to conclude that in both cases d(O0, O) < D'(L, C, X, X'), forcing 9 0 to be an isometry. [] 9. AN ABRIDGED VERSION OF THE ARGUMENT In this appendix we offer an introduction to the proof of Theorem 1.1.2 via the special case when X = X'= ~ � I'I ~. Step 1. ~ The structure of asymptotic cones r 2 � H2), z~). Readers unfamiliar with asymptotic cones should read section 2.4. By 2.4.4, any asymptotic cone r I-I ~, x~) is a CAT0r ) space for every !r so it is a metric tree; since there are large equilateral triangles centered at any point in I-I 2, the metric tree branches every- where. The ultralimit operation commutes with taking products, so one concludes that ~-lim(X,(I-lP X I-IP), z,) ___ ~-lim(X, H z, x~) X ~-lim(N. It~,y,) where z~ = x~ � y~ and X denotes the Euclidean product of metric spaces. So any asymptotic cone of ttP x I-IP is a product of metric trees which branch everywhere. Step 2. -- Planes in a product of metric trees are " locally finite ". For i = 1, 2 let T~ be a metric tree. For simplicity we assume that geodesic segments and rays are extendible RIGIDITY OF QUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 195 to complete geodesics. Since the convex hull of two geodesics in a metric tree is contained in the union of at most 3 geodesics, the convex hull of two 2-flats 3q x ~, C T, x T 2 is contained in at most nine 2-flats. Section 6 may now be read up to the paragraph after lemma 6.2. I, replacing the word " apartment " with " 2-flat ", and corollary 4.6.8 with the observation above. Hence every topologically embedded plane in T x x T~ is locally contained in a finite number of 2-flats. Step 3. -- Homeomorphisms of products of nondegenerate trees preserve the product structure. We now make the additional assumption that our metric trees T~ branch everywhere: for every x e T~, T~\x has at least 3 components. Let P C TI x T 2 be a topologically embedded plane, and let z----x x y e P. We know that there axe finite trees ~f~ C T~ with z eT a x TzC T1 X T, so that B,(r)r~ PC B,(r)n (T x x T2). Shrinking r if necessary, we may assume that T 1 and T2 are cones (x e T1 and y e T~ are the only vertices). Elementary topological arguments using local homology groups show that B,(r) r~ P coincides with B,(r) n (UQ,), where each Q~ C T1 � T2 is a quarter plane with vertex at z, i.e. a set of the form ~" x 8C Tx x T~ where -(C T 1 (resp. 8C T,) is a geodesic leaving x (resp. y). Say that two sets $1, S z C T~ � T 2 have the same germ at z if Sx r~ U ---= S 2 n U for some neighborhood U of z. We see from the above that for every z e P, P has the same germ at z as a finite union of quarter planes. Moreover, since the intersection of two quarter planes Q~, Q, with vertex at z either has the same germ as Q,, the same germ as a horizontal or vertical segment, or the same germ as { z }, it follows that a set S C T1 x T2 has the germ of a quarter plane with vertex at z if and only if it has the same germ as a two-dimensional intersection of topologically embedded planes, and is minimal among such. Hence we have a topological characterization of 2-flats and vertical/horizontal geodesics: a closed, topologically embedded plane P C T 1 x Ts is a 2-flat if for every z e P, P has the same germ at z a the union of four quarter planes with vertex at z; a closed connected subset S C Tx x T z is a vertical or horizontal geodesic if for every z e S, S has the same germ at z as the boundary of two adjacent quarter planes with vertex at z. From this one may easily recover the product structure on T x X T~ using only the topology of T x � T,. Hence a homeomorphism ~:Tx x T, ~ Tx x Ts preserves the product structure (although it may swap the factors, of course). Step g. ~ Q~asi-isometries of I-IP x I-lP preserve the product structure. Let ~ : I'l~ x II ~ --+112 x 1"12 be a quasi-isometry. If z, z' e I'P � 1-I 2, let O(z, z') be the angle between the segment zz' and the horizontal direction. Sublemma 9.0.12. -- There is a function f: [0, oo) --+R with lim,_.| = 0 so that if z, z' are horizontal, then [ 0(q)(z), q)(z')) -- ~/4 ] > :~/4 --f(r). 196 BRUCE KLEINER AND BERNHARD LEEB Proof. -- If not, we could find a sequence z,, z~ e H 2 X H 2 of horizontal pairs so that ~-~ : d(z~, z~) : ~ and lim sup~_~o [ 0(O(z~), O(z')) -- ~/4 ] < z~/4. Then z,o , z'~ ~ o-lim(~,(H 2 � H2), z,~) is a horizontal pair with 0(O,o(z,o), Oo(z'~)) ~= 0, n/2. This contradicts step 3. [] Since any two horizontal pairs z,, z~ and z~, z~ may be joined with a continuous family z,, z'~ of horizontal pairs with min d(z,, z',) >>. min(d(zx, z~), d(z~, z'2)), we see that for horizontal pairs z, z', the limit lima(,.,,~_~o o 0(O(z), O(z')) exists and is either 0 or ~/2. We assume without losing generality that the former holds. 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