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PUBLICATIONS MATHEMATIQUES DE L ' I .H .E .S

BRUCE KLEINER

BERNHARDLEEBRigidity of quasi-isometries for symmetric spaces and Euclidean buildings

Publications mathematiques de l'I.H.E.S., tome 86 (1997), p. 115-197.

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1.1. Back;

J) quasi-isometryX we have

L- 1 </(*!,*,)-

d(x',

Quasi-isometrie:e length spaces odiscontinuously

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 scalesthe metrics on X and X' by a factor X, then (L, C) quasi-isometries become (L, XG)quasi-isometries. Starting with a sequence Xj -> 0 we apply the ultralimit constructionof [DW, Gro] to take a limit of the sequence <5 : \ X ->• \ X', getting an (L, 0) quasi-isometry (i.e. a biLipschitz homeomorphism) Ow : Xw -*- X^ between the limit spaces.The first step is to determine the geometric structure of these limit spaces:

Theorem 1 . 2 . 1 . — The spaces XM 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 u , X^.We establish rigidity results for homeomorphisms of Euclidean buildings which aretopological analogs of Theorems 1.1.2 and 1.1.3:

Theorem 1.2 2. — Let Yi, Y4 be thick irreducible Euclidean buildings with topo-

logically transitive affine Weyl group (cf. section 4.1.1,), and let Y = E" X I I f = 1 Y 4 ,Y' = E"' X I l* ' = 1 Y;.. If Y : Y -> Y' is a homeomorphism, then n = n', k = k', and after

reindexing factors there are homeomorphisms Y 4 : Y , - > Y J so that p'o Y = Ux¥i o p where

p : Y -> Ilf= 1 Y4 and p': Y' -> JTjLi Y4 are the projections.

Theorem 1.2 3. — Let Y be an irreducible thick Euclidean building with topologicallytransitive affine Weyl group and rank > 2. Then any homeomorphism from Y to a Euclideanbuilding is a homothety.

For comparison we remark that if Y and Y' are thick irreducible Euclideanbuildings with crystallographic (i.e. discrete cocompact) affine Weyl group, then onecan use local homology groups to see that any homeomorphism carries simplices tosimplices. In particular, the homeomorphism induces an incidence preserving bijectionof 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 homeomorphismA -> 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*F : X -> X ' carries apartments to apartments.

In the third step, we deduce Theorems 1 1 2 and 1.1.3 from their topologicalanalogs. 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 O : X -> X ' is an (L, G) quasi-isometry, then the imageof a maximal flat in X under O lies within uniform Hausdorff distance of a maximal

118 BRUCE KLEINER AND BERNHARD LEEB

flat in X' ; the Hausdorff distance can be bounded uniformly by (L, G). In the caseof Theorem 1.1.2 we use this to deduce that the quasi-isometry respects the productstructure, and in the case of Theorem 1.1.3 we use it to show that O induces a well-defined homeomorphism SO : 8X. -> SX' of the geometric boundaries which is anisometry of Tits metrics. We conclude using Tits' work [Til] (as in [Mos]) that d0 : X -> X', and <2(O, O0) is bounded uniformly by (L, C).

The reader may wonder about the relation between Theorems 1.1.2 and 1.1.3and Mostow's argument in the higher rank case. An important step in Mostow's proofshows that if F acts discretely and cocompactly on symmetric spaces X and X', thenany F-equivariant quasi-isometry <J>: X -*• X' carries maximal flats in X to withinuniform distance of maximal flats in X'. The proof in [Mos] exploits the dense collectionof 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 ofmaximal flats in X' by any quasi-isometry (2); for instance, by analogy with the rank 1case one may ask whether any r-quasi-flat (3) in a symmetric space of rank r must liewithin bounded distance of a maximal flat. The answer is no. If X is a rank 2 symmetricspace, then the geodesic cone U s e s / t f over any embedded circle S in the Tits boundary8Tlta 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 letr = rank(X). Given L, G there are D, D ' e N such that every (L, G) r-quasi-flat Q,C X lieswithin the D-tubular neighborhood N D (Uj . 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 8Tlte X.

It follows easily that if L is sufficiently close to 1 (in terms of the geometry of thespherical Coxeter complex (S, W) associated to X) then any (L, C) r-quasi-flat in Xis 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 yearafter we had obtained the main results of this paper for symmetric spaces.

We would like to mention that related rigidity results for quasi-isometries havebeen proved in [Sch].

1.3. Organization 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-

(!) If Zr C F acts cocompactly on a maximal flat F C X, then Z' will stabilize O(F) and a flat F ' in X' .One can then get a uniform estimate on the Hausdorff distance between O(F) and F'.

(2) Obviously this statement is true by Theorems 1.1.2 and 1.1.3.(3) An r-quasi-flat is a quasi-isometric embedding cp : E r —> X ; a quasi-isometric embedding is a map

satisfying condition (1), but not necessarily (2).

mard spaces to the non locally compact case. This is needed when we study the limitspaces X u which are non locally compact Hadamard spaces.

Sections 3 and 4 give a self-contained exposition of the building theory usedelsewhere in the paper. This exposition has several aims. First, we hope that it will makebuilding theory more accessible to geometers since it is presented using the languageof 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 onmetric 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 (sphericaland Euclidean) arise as geometric limits of symmetric spaces, and we found that thegeometric 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 ourneeds.

Section 5 shows that the asymptotic cone of a symmetric space or Euclideanbuilding is a Euclidean building.

Section 6 discusses the topology of Euclidean buildings, proving Theorems 1.2.2,1.2.3, 1.2.4.

Section 7 proves that if X, X' and <5> are as in Theorem 1.1.2, then the imageof a maximal flat under <& is uniformly Hausdorff close to a flat (actually the hypotheseson X and X' can be weakened somewhat, see Corollary 7.1.5). General quasiflats arealso 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 thesymmetric space case.

1.4. Suggestions to the reader

Readers who are already familiar with building theory will probably find it usefulto 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' = H2 X H2 already containsmost of the conceptual difficulties of the general case, but one can understand theargument in this case with a minimum of background. To readers who are unfamiliarwith asymptotic cones, and readers who would like to quickly understand the proof ina special case, we recommend an abbreviated itinerary, see appendix 9. In general,when the burden of axioms and geometric minutae seems overwhelming, the readermay read with the Rank 1 x Rank 1 case in mind without losing much of the mathe-matical content.


CONTENTS

1. Introduction 115

1.1. Background and statement of results 1151.2. Commentary on the proof 1171.3. Organization of the paper 1181.4. Suggestions to the reader 119

2. Preliminaries 121

2.1. Spaces with curvature bounded above 1212.1.1. Definition 1212.1.2. Coning 1222.1.3. Angles and the space of directions of a CAT(ic)-space 123

2.2. CAT(l)-spaces 1242.2.1. Spherical join 1242.2.2. Convex subsets and their poles 125

2.3. Hadamard spaces 1252.3.1. The geometric boundary 1262.3.2. The Tits metric 1262.3.3. Convex subsets and parallel sets 1282.3.4. Products 1292.3.5. Induced isomorphisms of Tits boundaries 130

2.4. Ultralimits and asymptotic cones 1312.4.1. Ultrafilters and ultralimits 1312.4.2. Ultralimits of sequences of pointed metric spaces 1312.4.3. Asymptotic cones 133

3. Spherical buildings 134

3.1. Spherical Coxeter complexes 1343.2. Definition of spherical buildings 1353.3. Join products and decompositions 1363.4. Polyhedral structure 1383.5. Recognizing spherical buildings 1393.6. Local conicality, projectivity classes and spherical building structure on the spaces of directions 1403.7. Reducing to a thick building structure 1413.8. Combinatorial and geometric equivalences 1433.9. Geodesies, spheres, convex spherical subsets 1443.10. Convex sets and subbuildings 1443.11. Building morphisms 1453.12. Root groups and Moufang spherical buildings 147

4. Euclidean buildings 149

4.1. Definition of Euclidean buildings 1494.1.1. Euclidean Coxeter complexes 1494.1.2. The Euclidean building axioms 1504.1.3. Some immediate consequences of the axioms 151

4.2. Associated spherical building structures 1514.2.1. The Tits boundary 1514.2.2. The space of directions 152

4.3. Product (-decomposition) s 1534.4. The local behavior of Weyl-cones 154

4.4.1. Another building structure on Sp X, and the local behavior of Weyl sectors 155

4.5. Discrete Euclidean buildings 156

4.6. Flats and apartments 1574.7. Subbuildings 1594.8. Families of parallel flats 1604.9. Reducing to a thick Euclidean building structure 1614.10. 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 1675.2. Asymptotic cones of symmetric spaces are Euclidean buildings 169

6. The topology of Euclidean buildings 171

6.1. Straightening simplices 1726.2. The local structure of support sets 1736.3. The topological characterization of the link 1746.4. Rigidity of homeomorphisms 174

6.4.1. The induced action on links 1756.4.2. Preservation of flats 1756.4.3. Homeomorphisms preserve the product structure 1756.4.4. Homeomorphisms are homotheties in the irreducible higher rank case 1766.4.5. The case of Euclidean de Rham factors 177

7. Quasiflats in symmetric spaces and Euclidean buildings 177

7.1. Asymptotic apartments are close to apartments 1777.2. The structure of quasi-flats 180

8. Quasi-isometries of symmetric spaces and Euclidean buildings 185

8.1. Singular flats go close to singular flats 1858.2. Rigidity of product decomposition and Euclidean de Rham factors 1878.3. The irreducible case 187

8.3.1. Quasi-isometries are approximate homotheties 1888.3.2. Inducing isometries of ideal boundaries of symmetric spaces 1908.3.3. (1, A)-quasi-isometries between Euclidean buildings 191

9. An abridged version of the argument 194

REFERENCES 196

2. PRELIMINARIES

2.1. Spaces with curvature bounded above

General references for this section are [ABN, Ba, BGS].

2.1.1. Definition

If K e R, let M^ be the two-dimensional model space with constant curvature K ;let D(K) = Diam(M^). A complete metric space (X, | . |) is a CAT(ic)-space if

1. Every pair xx, x% e X with | xx x2 | < D(K) is joined by a geodesic segment.

2. Triangle or Distance Comparison.

Every geodesic triangle in X with perimeter < 2D(K) is at least as thin as thecorresponding triangle in M^. More precisely: for each geodesic triangle A in X withsides alt (T2, cr3 with Perimeter (A) = | a1 | + | <r2 | + | as | < 2D(K) we construct a

16

comparison triangle A in M£ with sides % satisfying | a4 | = | o |. Each point x on A

corresponds to a unique point jcon A which divides the corresponding side in the same

ratio. We require that for all x1} x2 e A we have | x1 x2 | ^ | Tcx x?2 |-

Remark 2 . 1 . 1 . — Note that we do not require X to be locally compact. Also, X need notbe path connected when K > 0. This is slightly more general than some other definitions in theliterature.

Example 2 . 1 . 2 . — A complete l-connected Riemannian manifold with sectional cur-vature < K < 0 and all its closed convex subsets are GKT{yi)-spaces.

In particular, Hadamard manifolds are CAT(0)-spaces. This is why we will alsocall CAT (0)-spaces Hadamard spaces.

Example 2 . 1 . 3 (Berestovski). — Any simplicial complex admits a piecewise sphericalGAT(l) metric.

Condition 2 implies that any two points xx, x2 with | xx x2 | < D(K) are connectedby precisely one geodesic; hence we may speak unambiguously of x1 x2 as the geodesicsegment joining x1 to x2. The CAT(K)-spaces for K < 0 are contractible geodesic spaces.

To see that upper curvature bounds behave well under limiting operations, it isconvenient to use an equivalent definition of CAT (K)-spaces which only refers to finiteconfigurations of points rather than geodesic triangles. If v, x,y,p e X, and~, %y,p e M£we say that 1", 7, y, 'p form a ^-comparison quadruple if

1. /T lies on % y.2 . \ \ v x \ - \ v x \ \ < 8 , \ \ v y \ - \ V y \ \ < * , \ \ x y \ - \ 7 y \ \ < 8 , \\ x P \ - \ x p \\ < Z,

\\py\-\py\\<$.By a compactness argument, we note that there exists a function 8K(P, s) > 0

such that for every s > 0, and every quadruple of points v, x, y, p in a CAT (K)-space Xsatisfying | vx | + | xy \ + |yv | satisfies | vp \ < | "v]> | + e. We will refer to this condition as the 8K-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 (K)-space if and only if it satisfies the SK-four-pointcondition and every pair of points y e X with | xy \ < D(K) has approximate midpoints,i.e. for every s' > 0 there is a m e X with | xm \, \ my | < | xy |/2 + s'. To see this, notethat in the presence of the SK-four-point condition approximate midpoints are close toone another, so one may produce a genuine midpoint by taking limits. By takingsuccessive midpoints, one can produce a geodesic segment.

2.1.2. Coning

Let S be a metric space with Diam(S) < n. The metric cone G(S) over 2 is definedas follows. The underlying set will be S X [0, oo) / ~ where ~ collapses S X { 0 } to

a point. Given vl3 v2 e 2, we consider embeddings p : { v1} v2 } X [0, oo) -> E2 such that| p(vi} t) = | / | and Zo(p(o1, ^ ) , p(o2, tf2)) = | vx v2 |, and we equip G(S) with the uniquemetric for which these embeddings are isometric. The space G(S) is CAT(O) if andonly if S is CAT(l).

2.1.3. Angles and the space of directions of a CAT(ic)-space

Henceforth we will say that a triple v, x, y defines a triangle A(z>, x,y) provided

| vx | + | xy | + |yv \ < 2 Diam(M^). The symbol Lv(x,y) will denote the angle of thecomparison triangle at the vertex ~. If x', y' are interior points on the segments Hx, vy,

then Z.v(x',y') < Av(x,y). From this monotonicity it follows that lira,, y,_v Lv{x',y')exists, and we denote it by Av(x,y). This definition of angle coincides with the notionof the angle between two segments in the Riemannian case. One checks that one obtainsthe same limit if only one of the points x', y' approaches v:

(3) Av(x,y) 2

Av satisfies the triangle inequality. Note that from the definition we have

(4) Av(x,y) < 2,{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 Av(x,y) = Lv[x,y),then also the other angles of the triangle A(», x,y) coincide with the corresponding comparison angles;moreover the region in M^ bounded by the comparison triangle can be isometrically embedded into Xso that corresponding vertices are identified.

The angles of a triangle depend upper-semicontinuously on the vertices:

Lemma 2 . 1 . 5 . — Suppose v, x,y e X define a triangle, v 4= x,y, and vk -> v, xk ->-#,

yk - > j . Then vk, xk, yk define a triangle for almost all k and

AVk(xk,yk) < Z.v(x,y).

In the special case that vke~vx~k — {v} holds \imk^.<a /LVk(xk,yk) = Z.v(x,y) and^ L n { v , y k ) = TC - L v ( x , y ) .

Proof. — For x' e~vx — { v } and y' e vy — { v } we can choose sequences of pointsw i t h 4 ->x> a n d y'u -^y'- T h e n At(**»A) < ^-vJA>y'k) -»• A(* '» / )

and the first assertion follows by letting x',y' ->• v. If vk evxk—{ v, xk} then

A(**>J*) < anglesum(A(», vk,yk)) - Z-Vk{v,yk) and n - Z.Vk{v,jk) ^ ZVk{xk,yk) whilelim sup anglesum(A(», vk,yk)) < n. Sending k to infinity, we get

A(*>J) < TC - lim inf /-Vk(v,yk) < lim inf Z.Vk{xk,yk)

and hence the second assertion. •

The condition that two geodesic segments with initial point v e X have anglezero at v is an equivalence relation; we denote the set of equivalence classes by S* X.The angle defines a metric on 2* X, and we let 2 , X be the completion of 2* X withrespect to this metric. We call elements of 2,, X directions at v (or simply directions),and vie denotes the direction represented by vx. We define the logarithm map as the maplog,, = logE x : B,,(D(K))\Z> -> 2 e X which carries x to the direction vx. The tangentcone of X at v, denoted Co X, is the metric cone C(28 X); we have a logarithm map

Given a basepoint v e X, X E X with d(v, x) < D(K), and X e [0, 1], let Xx e Xbe the point on ~vx satisfying | v(\x) |/| vx \ = X. We define a family of pseudo-metrics on B0(D(K)) by ds(x,y) = d(zx, sj>)/s. They converge to a limit pseudo-metric d0. The pseudo-metric space (B,,(D(K)), dt) satisfies the SEa K-four-point condition,so the limit pseudo-metric space (B,(D(K)) , d0) satisfies the 80-four-point condition. Butdo(x,y) = d(logv x, log,jy) where log,: B,(D(K)) -S- Gv X is the logarithm defined above,so we see that the tangent cone Cs X satisfies the 80-four-point condition (C(S* X)is dense in CB X, and every four-tuple in G(2* X) is homothetic to a four-tuple inlogr(Br(D(K))). If zx is the midpoint of the segment (Xx) (Xy), then

d{logv x, logoy) = lim - d(ex, zy)

( , e )

max lim U (log, *, - log, ztj, lim U (log, *, - log, zsj j .

50 Ge X also has approximate midpoints. Since C, X is complete, it is a CAT(0)-space;consequently 2S X is a CAT(l)-space. This fact is due to Nikolaev [Nik].

2.2. CAT(l)-spaces

GAT(l)-spaces are of special importance to us, because they will turn up as spacesof directions and Tits boundaries of Hadamard spaces.

2.2.1. Spherical join

Let Bx and B2 be GAT(l)-spaces with diameter Diam(B,) < 7t. Their sphericaljoin Bj o B2 is defined as follows. The underlying set will be Bt X [0,7t/2] X B2 /~, where" ~ " collapses the subsets { ^ 1 } x { 0 } x B a and BX X { reft } X { bz } to points. Givenbt, b\ eBj (i = 1, 2), we consider embeddings p : { b1} b[ } X [0, TC/2] X { b2, b'2 } -> S3.We think of S3 as the unit sphere in C2 and require that 11-> p(i l 51, b2) and t' H->- p(b[, t', b'2)are unit speed geodesic segments whose initial (resp. end) points lie on the great circle51 X { 0 } (resp. { 0 } X S1) and have distance d^(bx, b[) (resp. dBi(b2, b'2)). The distance

of the points in Bj o B2 represented by [bly t, bz) and (b[, t', b'2) is then denned as the(spherical) distance of their p-images in S3; it is independent of the choice of p. To seethat Bx o B2 is again a GAT(l)-space and that the spherical join operation is associative,observe that the metric cone C(Bi o B2) is canonically isometric to C(BX) X C(Ba) andthat the product of CAT (0)-spaces is GAT(O).

The metric suspension of a CAT(l)-space with diameter < n is denned as its sphericaljoin with the CAT(l)-space { south, north } consisting of two points with distance n.

Lemma 2 . 2 . 1 . — Let B x and B 2 be GAT (I)-spaces with diameter it and suppose s is anisometrically embedded unit sphere in the spherical join B = B 1 o B 2 . Then there are isometricallyembedded unit spheres si in Bi so that 5X o s2 contains s.

Proof. — We apply lemma 2.3.8 to the metric cone G(B) £ C(BX) X C(B2). Theset C(s) is a flat in G(B) and hence contained in the product of flats F4 s 0(6,),fj : = dTjtB Ft is a unit sphere in Bf and sx o s2 = dTite(Fi X F2) 2 dTlts C(s) = s. D

2.2.2. Convex subsets and their poles

We call a subset C of a CAT(l)-space B convex if and only if for any two pointsp, q e G of distance d(p, q) < it the unique geodesic segment/></ is contained in C. Closedconvex subsets of B are CAT(l)-spaces with respect to the subspace metric inheritedfrom B. Basic examples of convex subsets are tubular neighborhoods with radius ^ n/2of convex subsets, e.g. balls of radius < it/2.

Suppose that C C B is a closed convex subset with radius Rad(C) ^ iz, i.e. for eachp e C exists q e C with d(p, q) > it. We define the set of poles for C as

Poles(C):={,)eB:«*(i,, .)|o = !?)

If Diam(C) > iz then C has no pole. If Diam(C) = Rad(C) = TT then Poles(C) is closedand convex, because it can be written as an intersection Poles(C) = H ^ e c BJt/2(^) ofconvex balls. The convex hull of C and Poles (C) is the union of all segments joiningpoints in C to points in Poles(C), and is canonically isometric to CoPoles(C). Thisfollows, for instance, when one applies the discussion in section 2.3.3 to the parallelsets of C(C) in the metric cone G(B).

Consider the special case that G consists of two antipodes, i.e. points with distance TC,£ and %. Then the convex hull of { 5, % } and Poles ({ \, % }) is exactly the union of mini-mizing geodesic segments connecting E,, f and it is canonically isometric to the metricsuspension of Poles ({ \, % }).

2.3. Hadamard spaces

We will call CAT (0)-spaces also Hadamard spaces, because they are the syntheticanalog of (closed convex subsets in) Hadamard manifolds, i.e. simply connected completemanifolds of nonpositive curvature, cf. example 2.1.3.


2.3.1. The geometric boundary

Let X be a Hadamard space. Two geodesic rays are asymptotic if they remain atbounded distance from one another, i.e. if their Hausdorff distance is finite. Asympto-ticity is an equivalence relation, and we let dx X be the set of equivalence classes ofasymptotic rays; we sometimes refer to elements of dx X as ideal points or ideal boundarypoints. For any point x e X and any ideal boundary point ^ e ^ X there exists a uniqueray xE, starting at x which represents £. The pointed Hausdorff topology on rays emanatingfrom x e X induces a topology on 8n X. This topology does not depend on the basepoint x and is called the cone topology on S^X; d^ X with the cone topology is calledthe geometric boundary. The cone topology naturally extends t o X u ^ X. If X is locallycompact, then dx X and X ^ X u ^ X are compact and X is called the geometriccompactification of X.

2.3.2. The Tits metric

Earlier we defined the angle between two geodesies vx, vy at v e X by using the

monotonicity of comparison angles 2-v(x',y') as x' -> v, y' -> v. Now we consider a pair

of rays z>£, irt\, and define their Tits angle (or angle at infinity) by

(5) Z T i ^ , 7 ) ) : = ^ i m ^ 2 , , ( * ' , / )

where x' e v\ and y' e vrj. Z-Tits defines a metric on dx X which is independent of thebasepoint v chosen. We call the metric space dTltsX: = {dx X, Z-Titg) the Tits boundaryof X and Z_Titg the Tits (angle) metric. The estimate

*-

implies, combined with (4):

Consequently, the Tits angle can be expressed as

(6) Alta(5» l ) =

for any geodesic ray r: R+ ->• X asymptotic to ^ or TJ, and also as:

(7) ZTlte(^7))

Still another possibility (the last one which we will state) to define the Tits angle is asfollows: If rt: R + -> X are geodesic rays asymptotic to ££ then

w 2

The next lemma relates the cone topology on <?„ X to the Tits topology. Fix v e Xand consider the comparison angle

2v:(X\{v}) X (X\{*})->[0,7t].

By monotonicity, it can be extended to a function

2 , : ( X \ { * } ) X ( X \ { * } ) - [ 0 , T U ] .

Note that for l^ed^ X, we have LV{1, YJ) = Z.Tlta(5, YJ).

Lemma 2 . 3 . 1 (Semicontinuity of comparison angle). — r f o awg/tf Z e is lower semi-continuous with respect to the cone topology. If x k , y k , \, i\ e X — { v } are such that E, — l im^.^.^ xk

and 7) = l i m ^ ^ j j , then /.,(£, YJ) ^ liminfj^.^ /_v{xk,yk).

Proof. — We treat the case £, •/) e dx X, the other cases are similar or easier. Sincethe segments (or rays) vxk, vyk are converging to the rays p£, »rj respectively, we maychoose x'k e vxk andj^ e vy~ksuch that | x'k v \, |y'k v | -> oo and d(xk, vfy -> 0, d(y'k, vy) -> 0.Hence by triangle comparison we have

Lemma 2.3.2. — Every pair \, t\ e dM X w//A ZTits(£, YJ) < 7i to a midpoint.

Proof. — Pick j e X . Take sequences xi e p^, jy4 6 SYJ with | *i | = | J* | -»• °o. Let mi

be the midpoint of x~y{. Since A(», AJj, ) is isosceles, Z^^.m,) = Ztl(mi,jvj) < Ze(^,_>'j)/2,by lemma 2.3.1 it suffices to show that vm^ converges to a ray Ujx, for some [x e ^ X.

For i <j, set ^ : = | vxi |/| Wj |. By triangle comparison, we have the followinginequalities:

Since Xi3-. (| x^j |/| ^ j 4 |) ->• 1 as £,j -> oo, we have

Wi(\, mt)

and, since Z.Tits(i;, TJ) < it, this in turn implies:

Fixing t > 0, if we set t/\ vmi \ = S4, then | («, OT{) (», \ } ms) | -> 0 as !,_/ ~> oo. Since| ^(^ m4) | == t, this shows that the segments vmi converge in the pointed Hausdorfftopology to a ray U\L as desired. D

The completeness of X implies that (dm X, Z.Tlto) is complete. The metric coneG(8a} X, Z.Ttts) (the Tits cone) is complete and has midpoints. Moreover, since everyquadruple in C(doo X, Z.Tlta) is approximated metrically (up to rescaling) by quadruplesinX, C ^ X, Z.TitB) satisfies the S0-four-point condition and is therefore a GAT(0)-space.By section 2.1.1 we conclude:

Proposition 2.3.3. — The Tits boundary of a Hadamard space is a CAT(l)-space.

There is a natural 1-Lipschitz exponential map exp,, : C(#Tit8X) -> X defined asfollows: For [(£,*)] e C(dTiUs X) = dTm X X [0, oo)/~, let expp[(%, t)] be the point onpi, at distance t from p. The logarithm map logp :X—{/>}-> S P X extends continuouslyto the geometric boundary and induces there a 1-Lipschitz map logj,: dTitB X -> Sp X.The Triangle Filling Lemma 2.1.4 implies the following rigidity statement:

Flat Sector Lemma 2 . 3 . 4 . — Suppose the restriction of log,,: dTlte X ->- S p X to thesubset A £ 0Tito X is distance-preserving. Then the restriction of e x p p : C(STits X) -> Xto G(A) £ C(dTit8 X) u are 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 theunique geodesic segment connecting them. Closed convex subsets of Hadamard spacesare Hadamard themselves with respect to the subspace metric. Important examples ofconvex sets are tubular neighborhoods of convex sets and horoballs. We will denoteby HB5(x) the horoball centered at the point \ e dOT X and containing x e X in itsboundary.

Let Cx and C2 be closed convex subsets of a Hadamard space X. Then by (4),the distance function d(., C2)|c = dCi c : G1 —>-R 0 is convex and the nearest pointprojection wcJ^ : G1 -> C2 is distance-nonincreasing; dCi c is constant if and onlyif 7r0JCi is an isometric embedding. In this situation, we have the following rigiditystatement:

Flat Strip Lemma 2 3 .5 . — Let C1 and C2 be closed convex subsets in the Hadamard space X.

Ifdc | c = d then there exists an isometric embedding ty : Gx X [0, d] -> Xsuch that <\i(., o) = idc

and <\>{.,d) = TZC L .

This is easily derived from the Triangle Filling Lemma 2.1.4, respectively fromthe following direct consequence of it:

Flat Rectangle Lemma 2 . 3 . 6 . — Let xi e X, i e Z/4Z, be points so that for all i holdsA«J(*»- I» *» + i) ^ u /2 - Then there exists an embedding of the flat rectangular region[0, | x0 xt |] X [0, | A;X AT2 |] C E 2 into X carrying the vertices to the points x{.

We call the closed convex sets Gl9 C2 S X parallel, Gx || G2, if and only if </cJCl

and t/cjd a r e constant, or equivalently, Jtojci and 7rCl|c2 a r e 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 geodesies,because two such subsets are parallel if and only if they have finite Hausdorff distance.(A Hadamard space is said to have extendible geodesies if each geodesic segment is containedin a complete geodesic.)

Let Y c X be a closed convex subset with extendible geodesies. ThenRad(#Tits Y) = 7t. The parallel set PY of Y is defined as the union of all convex subsetsparallel to Y; PY is closed, convex and splits canonically as a metric product

(9) PY s Y x N y .

Here N y is a Hadamard space (not necessarily with extendible geodesies) and thesubsets Y X { pt } are the convex subsets parallel to Y. The cross-sections of PY orthogonalto these convex subsets can be constructed as intersections of horoballs:

(10) {y} x NY = PY n (1 HB5(j) VyeY.£ 6 « Y

Applying the Flat Sector Lemma 2.3 A one sees furthermore that dTito NY is canonicallyidentified with Poles (dTitB Y) C dTits X; dTita PY is the convex hull in dTlts X of dTlta Yand Poles (dTjts Y) and we have the canonical decomposition:

(11) 8Tlt8PY ^ a ^ Y o P o l e s ^ Y ) .

2.3.4. Products

The metric product of Hadamart spaces X4 is defined as usual using the Pythegoreanlaw. It is again Hadamard and its Tits boundary and spaces of directions decomposecanonically:

(12) 5Tits(xi X . . . X X J = 8Tite Xj o . . . o aTit8 X n ,

(13) ^....v&i X . . . x X J = ^ X , o . . . . S , X , .

Proposition 2 . 3 .7 . —• If X is a Hadamard space with extendible geodesies then all join

decompositions of 9TIt8 X are induced by product decompositions of X.

17

Proof. — Assume that the Tits boundary decomposes as a spherical joinTito X = Bx o B_ x and consider, for x e X and i — ± 1, the convex subsets

C4(x) : = n 5 e B . HB5(x) obtained from intersecting horoballs. Using extendibility ofgeodesies, i.e. Rad ZB X = TZ, one verifies that 9Tita Q = B4, C4 has extendible geodesiesand C±1(x) are orthogonal in the sense that ~EseGi(x) = Poles(L!B C_4(#)). Furthermoreany two sets C^x) and G_1(x') intersect in the point rcCi(B)(x') = to.,W- The assertionfollows by applying the Flat Rectangle Lemma 2.3.6. •

Lemma 2 . 3 .8 . — Let X x and X 2 be Hadamard spaces and suppose that F is aflat in the

product space X = X x X X 2 . Then there are flats F< £ X4 so that F1 X F2 2 F.

Proof. — Consider unit speed parametrizations c, c' : R -> F for two parallelgeodesies y> T' i n F. Then c t: = TZX. O C and c\ : = 7rXj. o c' are constant speed parame-trizations for geodesies y o y,' in X4. Since the distance functions d:=dx(c,c') and</. ; = dx.(ct, cl) are convex, satisfy d2 = d\ -f- </g and <f is constant, it follows that the dt

are constant, i.e. y» a n ( i Y< a r e parallel. Since this works for any pair of parallel geodesiescontained in F, it follows that TTX>. F is a flat in Fi. •

2.3.5. Induced isomorphisms of Tits boundaries

We now show that any (1, A)-quasi-isometric embedding of one Hadamard spaceinto another induces a well-defined topological embedding of geometric boundarieswhich preserves the Tits distance.

Proposition 2 . 3 . 9 . —• Let X x and X 2 be Hadamard spaces and suppose that O : Xx ->- X 2

is a (1, A)-quasi-isometric embedding. Then there is a unique extension : Xx -> X 2 such that

1. f(8a Xx) c 8a X2,2. $ is continuous at boundary points,

3. ^la^Xi *s a topological embedding which preserves the Tits distance.

We let 0 . 0 =

Proof. — We first observe that there is a function e(R) (depending on A but noton the spaces Xx and X2) with s(R) -> 0 as R -> oo such that if p, x,y e Xx andd(p, x), d{p,y) > R then

(14) | 2v{x,y) - Z^Mx), <b(y) | < s(R)).

Lemma 2 .3 .10 . — Suppose that xt is a sequence of points in X x which converges to a boundarypoint \x. Then O(^) e X 2 converges to a boundary point £2.

Proof of Lemma. —Pick a base point/*. There are points y{ e pxi such that d(p,yt) -> oo

and linij j ' ^ . , , , /-P(j i,J'j) = 0. By (14), the points 3>(ji) converge to a boundary point £2.Applying (14) again, we conclude that O(^) converges to £2 as well. •

Proof of Proposition continued. — From the previous lemma we see that if xt and x\are sequences in Xx converging to the same point in 8X Xx then the sequences O ^ )and O(#f) converge to the same point in 8K X2 . This allows us to extend 0 to a well-defined map O : Xj -^ X2.

We now prove that <b is continuous at every boundary point £. Let xi e ~KX bea sequence of points converging to \ e dm Xx . By the lemma, we may choose y{ e Xx

withjj epxi so that for every R the Hausdorff distance between O(/>)and <b(p) Ofo) n BK((/>)) tends to zero as R ->• oo. Hence

by the lemma.

Another consequence of the lemma is that the image ray <&(/>£) diverges sublinearily

from the ray O(p) <&(£) in the sense that

Jlim — .dL(<J>(0£ n BK(/>)), <p(/>) ^(5) ^ BK(0(/>))) == ^

R-> oo R

where rfH denotes the Hausdorff distance. This implies that dx <1> = O \da> x preservesthe 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 ultralimits

Definition 2 . 4 . 1 . — A nonprincipal ultrafilter is a finitely additive probability measure won the subsets of the natural numbers N such that

1. co(S) = 0 or 1 for every S C N.2. w(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 elementw-lim a e X such that for every neighborhood U of w-lim a, a~J(U) C N has full measure.In particular, given any bounded sequence a : N -> R, co-lim a (or aa) is a limit pointselected by «.

2.4.2. Ultralimits of sequences of pointed metric spaces

Let (Xj, «Jj,*j) be a sequence of metric spaces with basepoints *4. ConsiderX,,, = {# e I I i 6 N Xj | </4(*4, *4) is bounded}. Since di(xi,yi) is a bounded sequencewe may define da '• X^ X X^ -> R by d^{x,y) = w-lim ^ ( x ^ j j ; ^ is a pseudo-distance. We define the ultralimit of the sequence (Xi5 di} *4) to be the quotient metricspace (Xu , da), xa e Xu denotes the element corresponding to * = (xt) e X B .*w : = (*«) is the basepoint of (Xw, rfu).

Lemma 2 . 4 . 2 . — If ( X o du *4) is a sequence of pointed metric spaces, then (Xw , da, * Jis complete.

Proof. — Let x'a be a Cauchy sequence in Xm, where x'a = to-lim x{. Let Nx = N.Inductively, there is an co-full measure subset N3- s N3- _ 1 such that i e N3- implies| 4(*f, *]) - <,(**, *') | < 1/2% for 1 < k, I < J. For i e N, - N , . ^ , define j>4 = *?'.Then xs

a -+ya. D

The concept of ultralimits is an extension of Hausdorff limits.

Lemma 2 . 4 . 3 . — If ( X o d%, *4) form a Hausdorff precompact family of pointed metric

spaces, then (XB , da, * u ) is a limit point of the sequence (X4, dt, *4) with respect to the pointed

Hausdorff topology.

Proof. — To see this, pick s, R, and note that there is an N such that we can findan N element sequence { x{ }f = j C X , which is e-dense in X{. The N sequences x{ for1 < J < N give us N elements in j j . e X , , If ya e Xw, ya e B*JR), then for w-a.e.(that is, co-almost every) i, di(yi,*i)<K. Consider da(yu, #£>). Given s > 0,I da(j>a,x'J — d^y^xl) | < s for to-a.e. i, which implies that flfu (j>a, x'J < s forsome 1 < J < N. Hence we have seen that B*(0(R) is totally bounded, and for all e > 0there is an e-net in B.m(R) which is a Hausdorff limit point of e-nets in the X/s. Itfollows that (Xj, di; *4) subconverges to (Xw, du, *M) in the pointed HausdorfF topology. •

In general, the ultralimit XM is not Hausdorff close to the spaces X< in the " approxi-mating " sequence. However, the Hausdorff limits of any precompact sequence ofsubspaces Y4 C X ; canonically embed into X u .

The importance of ultralimits for the study of the large-scale geometry derivesfrom the following fact: If for each i, f : X< -> Y4 is a (L, C)-quasi-isometry withdi(fi(*i)>*i) bounded, then t h e ^ induce an (L, C)-quasi-isometryi/(O: Xw -^-YM.

It follows that if for each i, and every pair of points ai,bi e X4 the distance d^, bt)is the infimum of lengths of paths joining a{ to b{ then every pair of points au, ba e XM

is joined by a geodesic segment.

Lemma 2 . 4 . 4 . — If (X4, d{, *{) is a CAT(tzj-space for each i, then so is (Xm , du, *a).

If da(aa, ba) < D ( K ) , then the geodesic segment aaba is an ultralimit of geodesic segments.

IfK^O and each Xj has extendible geodesies then each ray (respectively complete geodesic) in Xw

is an ultralimit of rays (respectively complete geodesies) in the X.t's.

Proof. — If each (Xj, dx, *4) is a CAT(K) length space, then clearly (Xtt) da, *w)satisfies the SK-four-point condition since this is a closed condition. Hence (XM, da, *M)is a CAT(K) length space since it is a geodesic space satisfying the SK-four-point condition.

If aa, ba e X a with | aa ba | < D(K), then there is a unique geodesic segmentjoining au to b^. On the other hand, if aa = w-lima4, b^ = w-lim bi} then the ultra-limit of the geodesic segments ai bt is such a geodesic segment.

RIGIDITY OF Q.UASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 133

Now suppose a0^, a^, . . . determine a ray, in the sense that

<«, «*) = 4X> <) + d»&, <£) for * < j< *•Let Nx = N. Inductively, there is an co-full measure N, £ N i _ 1 such that a° a\ is

within a 1/2'-neighborhood of the segment a\ a\ for i e N3-, 0 ^ l<j. For » e N3- — N3_1

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 geodesies follows from

similar reasoning. •

Lemma 2 . 4 . 5 . — Suppose that there is a D > 0 such that for each i, Isom(X i) has anorbit which is D-dense in X4 . If \ > 0 and X4 -> 0, then the ultralimit of (Xj, \ di} *4) eVindependent of the choice of basepoints *{ , and has a transitive isometry group.

2.4.3. Asymptotic cones

Let X be a metric space and let *„ e X be a sequence of basepoints. We definethe asymptotic cone Cone(X) of X with respect to the non-principal ultrafilter to, thesequence of scale factors Xn with to-lim Xn = oo and basepoints *„, as the ultralimitof the sequence of rescaled spaces (XB, dn, *„) : = (X, ~k~x-d, *„). When the sequence•*„ = * is constant, then Cone(X) does not depend on the basepoint * and has acanonical basepoint *w which is represented by any sequence (xn) C X satisfying<o-limn X"1. «"(*„, *) = 0, for instance, by any constant sequence (x).

Proposition 2.4.6. —•

• If X is a geodesic metric space, then Cone(X) is a geodesic metric space.• If X is a Hadamard space, then Gone(X) is a Hadamard space.• IfKisa CAT'(K)-space for some K < 0, then Cone(X) is a metric tree.• If the orbits o/"Isom(X) are at bounded Hausdorff distance from X, then Cone(X) is

a homogeneous metric space.• A (L, C) quasi-isometry of metric spaces 9 : X -> Y induces a bilipschitz map

Cone(<p) : Cone(X) -> Cone(Y) of asymptotic cones.

Assume now that X is a Hadamard space. Let (FJ n e N be a sequence of ^-flatsin X and suppose that to-limn X~1.a'(Fn, *) < 00. Then the ultralimit of the embeddingsof pointed metric spaces

1 . , A („ 1 ,

is a &-flatR* €_• Cone(X)

in the asymptotic cone. We denote the family of all A-flats in Cone(X) arising in thisway by


3. SPHERICAL BUILDINGS

Our viewpoint on spherical buildings is slightly different from the standard one:for us a spherical building is a GAT(l) space equipped with an extra structure. Thisviewpoint is well adapted to the needs of this paper, because the spherical buildingswhich 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 containanything new; the same results and many more can be found (often in slightly differentform) 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 isometrywhose fixed point set, its wall, is a subsphere of codimension one. If W C Isom(S) isa finite subgroup generated by reflections, we call the pair (S, W) a spherical Coxetercomplex and W its Weyl group.

The finite collection of walls belonging to reflections in W divide S into isometricopen 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. finiteintersections of hemispheres. A face of a chamber is an intersection of the chamber withsome walls.

A face (resp. open face) of S is a face (resp. open face) of a chamber of S. Two facesof S are opposite or antipodal if they are exchanged by the canonical involution of S; twofaces 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-apartmentor root is a hemisphere bounded by a wall and a regular point in S is an interior pointof a chamber. The regular points form a dense subset. The orbit space

Am o d :=S/W

with the orbital distance metric is a spherical polyhedron isometric to each chamber.The quotient map

(15) 6 = 9 s : S ^ A m o d

is 1-Lipschitz and its restriction to each chamber is distance-preserving. For 8, §' e Amod,we set

D(S, » ' ) : = { dB{x, *') | x, x' e S, 6* = S, 6*' = 8' }

and D+(8) : = D(8, 8)\{ 0 }.

Note that D+ is continuous on each open face of AmQd.An isomorphism of spherical Coxeter complexes (S, W), (S', W ) is an isometry

a : S -> S' carrying W to W . We have an exact sequence

1 -> W -> Aut(S, W) -> Isom(Amod) -> 1.

RIGIDITY OF QjUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 135

Lemma 3 . 1 . 1 . — If g e W, then Fix(^) c 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. Thereforethe stabiliser of a face cr C S fixes a pointwise. So for all g e W, FixQj) is a subsphereand a subcomplex, i.e. it is a singular sphere.

By the above, without loss of generality we may assume that Z is a singularsphere. Let Wz be the group generated by reflections fixing Z pointwise. If a is atop-dimensional face of the singular sphere Z then each W-chamber containing a iscontained in a unique Wz-chamber; therefore Wz acts (simply) transitively on theW-chambers containing a. Since W acts simply (transitively) on W-chambers, it followsthat Fixator(Z) = Fixator(a) = Wz . •

3.2. Definition of spherical buildings

Let (S, W) be a spherical Coxeter complex. A spherical building modelled on (S, W)is a CAT(l)-space B together with a collection J / of isometric embeddings i : S -^-B,called charts, which satisfies properties SB 1-2 described below and which is closed underprecomposition with isometries in W. An apartment in B is the image of a chart i : S -> B;i is a chart of the apartment i(S). The collection J / is called the atlas of the sphericalbuilding.

SB1. Plenty of apartments. — Any two points in B are contained in a commonapartment.

Let iA , iAs be charts for apartments Al9 A2, and let C = A± n A2, C = ^(G) C S.The charts iA. are ^-compatible if iA * o i^ c. is the restriction of an isometry in W.

SB2. Compatible apartments. — The charts are W-compatible.It will be a consequence of corollary 3.9.2 below that the atlas s/ 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 objectsin the spherical Coxeter complex. The building is called thick if each wall belongs toat least 3 half-apartments. The axioms yield a well-defined 1-Lipschitz anisotropy map (-1)

(16) 6 B : B - > S / W = : A m o d

satisfying the discreteness condition:

(17) dB(Xl, x2) e D(6B(*1), 8B(*,)) V x1} x2 e B.

If a : S ->- S is an automorphism of the spherical Coxeter complex, then we modifythe atlas by precomposing with a; the atlases obtained this way correspond to symmetries

(*) The motivation for this terminology comes from the role 8B plays in the structure of symmetric spacesand Euclidean buildings.

If s/' is an atlas of charts i: S' -> B giving a (S', W ) building structure on B,then this spherical building is equivalent to (B, si) if there is an isomorphism of sphericalCoxeter complexes a : (S', W) -* (S, W) so that s/' = { i o a | i e si }.

If B and B' are spherical buildings modelled on a Goxeter complex (S, W), withatlases s/ and s/', an isomorphism is an isometry B' such that the correspondencei H i p o i defines a bijection s/ -> s/'.

3.3. Join products and decompositions

Let B4, i = 1 , . . . ,» , be spherical buildings modelled on spherical Coxetercomplexes (Si5 W{) with atlases s/t and spherical model polyhedra A j ^ . ThenW : = Wx X . . . X Wn acts canonically as a reflection group on the sphereS = Sxo . . . o S , . We call the Goxeter complex (S, W) the spherical join of the Goxetercomplexes (So W4) and write

(18) (S,.W) = ( S 1 , W 1 ) o . . . o ( S . , W J .

The model polyhedron Amod of (S, W) decomposes canonically as

(19) Amoa = Aj a o d° . . .oA» o d .

The CAT(l)-space

(20) B = B 1 o . . . o B B

carries a natural spherical building structure modelled on (S, W). The charts i for itsatlas s/ are the spherical joins i = ix o . . . o tn of charts it e s/t, We call B equippedwith this building structure the spherical (building) join of the buildings B4.

Proposition 3 . 3 . 1 . — Let B be a spherical building modelled on the Coxeter complex (S, W)with atlas srf 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

2. There is a decomposition (20) ofHas a join of spherical buildings so that Bt = ^

Proof. — 1. We identify Amod with a W-chamber in S and define Sj to be theminimal geodesic subsphere containing A ^ . Then St c Poles(S,) for all i^j andhence S = Sx o . . . o Sn for dimension reasons. Each wall containing a codimension-oneface of Aj,,^ is orthogonal to one of the spheres S{ and contains the others. HenceW = Wx X . . . X Wn where W4 is generated by the reflections in W at walls orthogonalto S4. The group W4 acts as a reflection group on S4 and the claim follows.

2. Since any two points in B are contained in an apartment, one sees by applyingthe first assertion that the B4 are convex subsets and B is canonically isometric to the

join of GAT(l)-spaces B = Bx o . . . o BB. The collection of charts i |Si, i e stf, formsan atlas for a spherical building structure on B4 and B is canonically isomorphic to thespherical building join of the B4. D

We call a spherical polyhedron irreducible if it is a spherical simplex with diameter < n/2and dihedral angles < TT/2 or if it is a sphere or a point. Accordingly, we call a sphericalCoxeter complex (x) or a spherical building irreducible if its model polyhedron is irreducible.The spherical model polyhedron Amod has dihedral angles ^ TC/2. A polyhedron of thissort has a unique minimal decomposition as the spherical join (19) of irreducible sphericalsimplices (which may be single points) and, if non-empty, the unique maximal unitsphere contained in A,^ . By Proposition 3.3.1, (19) corresponds to unique minimaldecompositions (18) of the Coxeter complex (S, W) as a join of Coxeter complexesand (20) of B as a spherical building join. We call these decompositions the de Rhamdecompositions of (S, W) and B. The sphere factor in (19) occurs if and only if the fixedpoint set of the Weyl group is non-empty. We call the corresponding factor in the deRham decomposition the spherical de Rham factor.

If W acts without fixed point, then Amod is a spherical simplex (2) and the collectionof chambers in S and B give rise to simplicial complexes.

Lemma 3 . 3 . 2 . — Let (S, W) be an irreducible spherical Coxeter complex with non-trivial

Weyl group W. Then for each chamber a there is a wall which is disjoint from the closure a.

Proof. — Let T' be a wall and p e S be a point at maximal distance n/2 from T'.Pick a chamber a' containing/) in its closure. Then ? n T' = 0, because Diam(<j') < TC/2due to irreducibility. Since W acts transitively on chambers, the claim follows. •

Proposition 3 . 3 . 3 . — Assume that B± and B2 are CAT(l)-spaces and that their joinB = BJL o B2 admits a spherical building structure. Then the B4 inherit natural spherical buildingstructures from B. In particular, the spherical building B cannot be thick irreducible with non-trivialWeyl group.

Proof. — Applying lemma 2.2.1 to apartments in B, we see that there exista\, d2 e N so that every apartment A s B splits as A = A± o A2 where Aj is a </rdimensionalunit sphere in B4. Fix a chart i0 in the atlas s/ for the given spherical building structureon B. Denote by S2 the *4-sphere i^x B2 in the model Coxeter complex (S, W) and bySx : = Poles (S2) the complementary ^-sphere. The subgroup Wx £ W generated byreflections at walls containing S2 acts as a reflection group on S±. Consider all chartsi e s# with i |S2 = i0 |g2. The collection s/± of their restrictions i |s forms an atlas fora spherical building structure on B1 with model Coxeter complex (Sx,

(x) This definition is slightly different form the usual one, which corresponds to irreducibility of linearrepresentations.

(a) By [GrBe] [theorem4.2.4], AmOd " a simplex if W acts fixed point freely. Observe that having distanceless than TT/2 is an equivalence relation on the vertices. This implies the decomposition (19).

If B is thick, then its chambers are precisely the (closures of the) connectedcomponents of the subset of manifold points. Hence the joins a2 o a2 of chambers at C B(

are contained in chambers of B. So the chambers of B have diameter ^ n/2 and B cannotbe irreducible with non-trivial Weyl group. •

3.4. Polyhedral structure

Let A' be a face of A,,^ and let a : A' -»• B be the chart for a face in B, i.e. anisometric embedding so that GB o a = id |A,.

Sublemma 3 . 4 . 1 . — The image a(Int A') is an open subset O / "0B X (A ' ) .

Proof. — Let A; be a point in <r(Int A') and assume that there exists a sequence (xn)in 631(A')\c(Int(A')) which converges to x. There are points x'n elm(a) with®B(*») — BOO- Since 0B has Iipschitz constant 1 and a is distance-preserving, we have

dB(xn, x) 3* dAmod(QB(xn), 6B(*)) = dB(x'n, x)

and by the triangle inequality2 - ^ B ( * ^ ) > dB{xn, xn) > D+(6B(*J).

-*0

Since D + is continuous on Int A', the right-hand side has a positive limit:

l imD+(0 B (* B ) )=D + (e B (*) )>0 ,

a contradiction. D

Lemma 3 . 4 . 2 . — Any two faces of B with a common interior point coincide. Consequently,

the intersection of faces in B is a face in B.

Proof. — To verify the first assertion, consider two face charts ax , <r2 : A' -> B ofthe same type. By Sublemma 3.4.1, { 8 e A' | aj.(&) = <r2(8) } n Int A' is an open subsetof Int A'. It is also closed, and hence empty or all of Int A' if A' is connected. If A' isdisconnected, it must be the maximal sphere factor of Amod and all apartment chartsagree on A'. Hence CTX A, = a2 A, 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 underthe binary operations of union and intersection; we will denote this lattice by JfB. Inthe case that the Weyl group acts without fixed point, the chambers of B are simplices,and JTB is the lattice of finite subcomplexes of a simplicial complex. In general thepolyhedron of this simplicial complex is not homeomorphic to B since it has the weaktopology.


3.5. Recognizing spherical buildings

The following proposition gives an easily verified criterion for the existence of aspherical building structure on a CAT(l)-space.

Proposition 3 . 5 . 1 . — Let (S, W) be a spherical Coxeter complex, and let B be a CAT( l ) -space of diameter n equipped with a l-Lipsckitz anisotropy map 6B as in (16) satisfying thediscreteness condition (17). Suppose moreover that each point and each pair of antipodal regularpoints is contained in a subset isometric to S. Then there is a unique atlas s/ of charts i: S -> Bforming a spherical building structure on B modelled on (S, W), with associated anisotropy map 0B .

Proof — The discreteness condition (17) implies that, for any face A' of Amod,the restriction of 0B to Og^Int A') is locally distance-preserving and distance-preservingon minimizing geodesic segments contained in 0B"1(IntA'). Therefore, if A C B is asubset isometric to S, the restriction of 0B to AreB : = A n ©^(Int Amod) is locallyisometric and the components of AKg are open convex polyhedra which project via 0B

isometrically onto IntAmod. (17) implies moreover that Areg is dense in A. Hence Ais tesselated by isometric copies of Amod and there is an isometry iA with 0B o iA = 0S

which is unique up to precomposition with elements in W. If Ax and A2 are subsetsisometric to S, and iA , iA : S -> B are isometries as above then Ax n A2 is convex,and we see that iA and tA are W-compatible. We now refer to the isometries iA : S -> Bas charts and to their images as apartments. The collection s/ of all charts will be theatlas for our spherical building structure.

Since any point lies in some apartment, it lies in particular in a face, i.e. in theimage of an isometric embedding a : A' -> B of a face A' s Amod satisfying 0B o a = id |A,.Lemma 3.4.2 applies and the 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 7u lie in a commonapartment. It suffices to check this for any regular points xls x2, since any point liesin a chamber and an apartment containing an interior point of a chamber containsthe whole chamber (lemma 3.4.2). There is an apartment A1 containing xx. Considera minimizing geodesic c joining xx and x2. By sublemma 3.4.1, Ax is a neighborhoodof xx. Hence near its endpoint xx, c is a geodesic in the sphere Ax. Since B is a CAT(l)-space, we can extend c beyond xx inside At to a minimizing geodesic c of length njoining x2 through xx to a point x2 e Ax. By our assumption, the points x2, x2 are containedin an apartment A2, which contains all minimizing geodesies connecting x2 and x2,because x2 is regular. In particular c and therefore both points x1} x2 lie in A2. D

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 £ B be a subset

isometric to the Euclidean unit sphere of dimension d. Then T is an apartment in B.

3.6. Local conicality, projectivity classesand spherical building structure on the spaces of directions

Suppose that the spherical building B has dimension at least 1.

Lemma 3 . 6 . 1 . — Let (B, 0B) be a spherical building modelled on Amod, and let p,p e Bbe antipodal points, i.e. d(p,p) = iz. Then the union of the geodesic segments of length -K from p

to p is a metric suspension which contains a neighborhood of {p,p}.

Proof. — By the discussion in section 2 .2 .1 , the union of the geodesic segmentsof length 7c from p to p is a metric suspension. By (17) we can choose p > 0 such that{ q e B2p(/>) | 6B(y) = %{p) } = {p},Ue B2p(p) | 6B(?) = QB(p) } = {p}. If q <= Bp(£),then any extension of pq to a segment pqr of length n will satisfy 6(r) = 6(^), forcing r = pby the choice of p. Likewise, if we extend pq to a segment of length n, where q e Bp(j>),then it will terminate at p. Hence the lemma. D

As a consequence, for sufficiently small positive s, the ball Bs(p) is canonicallyisometric to a truncated spherical cone of height s over T,p B, the isometry given bythe " logarithm map " at p. In particular, 2* B = Sj, B. Any face intersecting Be(/»)contains p and the face o^ spanned by p.

The lemma implies furthermore that for any pair of antipodes p, p e B there isa canonical isometry

(21)

determined by the property that all geodesies c of length TC joining p and p satisfy

Two points in B are antipodal if and only if they have distance iz. Two faces ax

and «72 are antipodal or opposite if there are antipodal points \x and \% so that ^ lies inthe interior of o ; in this case each point in ox has a unique antipode in a2.

Definition 3 . 6 . 2 . — The relation of being antipodal generates an equivalence relation andwe call the equivalence classes projectivity classes.

Lemma 3 . 6 . 3 . — Suppose that the spherical building B is thick. Then every projectivity

class intersects every chamber.

Proof. — Let Cx and C2 be adjacent chambers, i.e. it = Cx n C2 is a panel. Itsuffices to show that for each point in Cl7 C2 contains a point in the same projectivityclass. To see this, pick an apartment A 3 Cx u C2 and let ft be the panel in A oppositeto 7t (ft = 7v is possible). Since B is thick there is a chamber C with G n A = ft. ThenG is opposite to both Gx and G2 and our claim follows. •

Pick p0 e S so that 0S(/><>) = QB(P)' Now consider the collection of all apart-ment charts iA: S -> B where iA(p0)

= P- These induce isometric embeddings2J>0

1A : S D 0

s -* 2 D B- Let WPo S Isom(SJ)o S) be the finite group generated by the

reflections in walls passing through p0-

RIGIDITY OF QJUASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 141

Proposition 3 . 6 . 4 . — The space 2P B together with the collection of embeddingsIiVoiA:T,1>oS - > S p B as above is a spherical building modelled on ( S ^ S , W J . If p e B isan antipode ofB, then we have a 1-1 correspondence between apartments (respectively half-apartments)in B containing {p,p} and apartments (respectively half-apartments) in 2 P B; 2 p B is thickprovided B is thick.

Proof. — Any two points pqx, pq2 e Sj, B lie in an apartment; namely choose q1} q2

close to p, then any apartment A containing qx, q2 will contain p and pqt e Sp A. SoSB1 holds. The space Sj, B satisfies SB2 since we are only using charts iA : S -*- B withiA(j>o) = ^ a n d B itself satisfies SB2. The remaining assertions follow immediately fromthe definition of the spherical building structure on Sp B. •

3.7. Reducing to a thick building structure

A reduction of the spherical building structure on B consists of a reflection subgroupW C W and a subset jtf' C s/ which defines a spherical building structure modelledon (S, W ) . The Aj^-direction map 0B can then be factored as TC o 6B where

is the A^-direction map for the building modelled on (S, W ) , and

is the canonical surjection.

Proposition 3 . 7 . 1 . — Let B be a spherical building modelled on the spherical Coxetercomplex (S, W), with anisotropy polyhedron Amod = W \ S . Then there exists a reduction ( W , s/')which is a thick building structure on B; W is unique up to conjugacy in W and si' is determinedby W . In particular, the thick reduction is unique up to equivalence, so the polyhedral structure isdefined by the GAT(l) space itself.

The proof will occupy the remainder of this paper.We set d = dim(B), RB = {p eB | Sj, B is isometric to a standard S*"1}, and

SB — B\RB . If p e B and p > 0 is small enough so that Bp(/>) is a (spherical) conicalneighborhood of p, then SB nBp(/>)\{/>} corresponds to the cone over S s B . It thenfollows by induction on dim(B) that SB n A is a union of Amod-walls for each apart-ment A C B.

Consider an apartment A C B, and a pair of walls Hj , H2 C A contained in SB.

Lemma 3 . 7 . 2 . •—• If Hg is the image of H 2 under reflection in the wall H x (inside the

apartment A), then H'2 is contained in S B .

Proof. — To see this, consider an interior point p of a codimension 2 face a ofHx n H 2 . The space S s B decomposes as a metric join ~Lp a o Bp where Bp is a 1-dimen-


sional spherical building, and the walls H1} H2, and W2 correspond to walls H1} H2,and H2 in B^; A corresponds to an apartment A in JSp. The wall H t is just a pair ofpoints in Bp, and this pair of points is joined by at least three different semi-circles oflength 7t. These three semi-circles can be glued in pairs to form three different apartmentsin Bp. Using the fact that an antipode of a point in SB also lies in SB , it is clear thatthe image of H2 under reflection in Hx is also in SB . Hence the wall S,, H'2 C Ep B iscontained in three half-apartments, and proposition 3.6.4 then implies that H'2 liesin three half-apartments. •

The reflections in the walls in A n SB generate a group GA, and by [Hum, p. 24]the only reflections in GA are reflections in walls in A n SB; also, the closures of connectedcomponents of A\S B are fundamental domains for the action of GA on A.

Sublemma 3 . 7 . 3 . — Let U s B be a connected component o /B \S B , and suppose U n A 4= 0for some apartment A. Then U £ A.

Proof. — The set U n A is open and closed in U, so U n A = U. •We claim that the isomorphism class of GA is independent of A. To show this,

it suffices to show that the isometry type of a chamber A£od is independent of A. Fori = 1, 2 let Aj be an apartment, and let A^d be a chamber for GA.. If A8C B is anapartment containing an interior point from each A£'od, then the sublemma givesA£j,,jC A3. But then the A^d are both chambers for GAj, so they are isometric. Henceeach pair (A, GA) is isomorphic to a fixed spherical Goxeter complex (S, Wth) for somereflection subgroup Wth £ W. We denote the quotient map and model polyhedron by

We call the closure of components of B\SB , Aôd-chambers. We can identify the^mod-chambers with Aôd in a consistent way by the following construction: Let A o s Bbe an apartment and p0 e Ao n RB be a smooth point. We define the retraction p : B -> Ao

by assigning to each point/) in the open ball Bn(p0) the unique point p(/>) e Ao for which

the segments pop and p0 p(p) have same length and direction pop = p0 p(p) a.tp0. Themap p extends continuously to the discrete set B\Bn(/0) which maps to the antipodeof />„ in Ao. If A is an apartment passing through p0 then A n Ao contains the A*^-chamber spanned by />„ and p|A : A ->• Ao is an isometry which preserves the tesselationsby chambers. Composing p with the quotient map Ao ->• Ao/GAo we obtain a 1-Lipschitzmap

(22) 6£:BÂ£ ld

which restricts to an isometry on each chamber. Applying proposition 3.5.1 we seethat B is a spherical building modelled on (S, W th); B is thick since we already verifiedin lemma 3.7.2 above that if H C SB is a wall, then it lies in at least three half-apartments.

Corollary 3 . 7 . 4 . — For i — 1, 2 let B4 be a thick spherical building modelled on (St, W4)with atlas stfi. If B2 is an isometry then we may identify the spherical Coxeter complexes

by an isometry a.: (S l 5 Wx) -> (S2 , W2) so that cp becomes an isomorphism of spherical buildings.

3.8. Combinatorial and geometric equivalences

We recall (section 3.4) that for any building B, JTB is the lattice of finite unionsof faces of B.

Proposition 3 . 8 . 1 . — Let B l 9 B2 be spherical buildings of equal dimension. Then any latticeisomorphism JTBX ->JfB2 is induced by an isometry Bx ->-B2 of CAT(l)-spaces. This isometryis unique if the buildings Bt do not have a spherical de Rham factor.

Proof. — First recall that lattice isomorphisms preserve the partial ordering byinclusion since Cx C C2 o C1 u C2 = C2.

We first assume that the buildings Bt have no de Rham factor and hence the DfBi

come from simplicial complexes. In this case the lattice isomorphism JTBX -> X~B2 carries^-dimensional faces of Bx to ^-dimensional faces of B2. To see this, note that vertices of B;are the minimal elements of the lattice Jf Bt and &-simplices are characterized (induc-tively) as precisely those subcomplexes which contain k + 1 vertices and are not containedin the union of lower dimensional simplices.

Consider a codimension-2 face a of a chamber C in B4. For an interior point sea,2 , B4 is isometric to the metric join S ( a o B? where B° is a 1-dimensional sphericalbuilding. The dihedral angle of C along <r equals the length of a chamber in the1-dimensional building B°.

Sublemma 3 . 8 . 2 . — The chamber length of a l-dimensional spherical building is determined

combinatorially as 2TZ/1 where I is the combinatorial length of a minimal circuit.

Proof. — Combinatorial paths in a l-dimensional spherical building determinegeodesies. Closed geodesies in a GAT(l)-space have length at least 27i since points atdistance < n are joined by a unique geodesic segment. The closed paths of length 2TCare the apartments. D

Proof of Proposition 3.8.1 continued. — As a consequence of the sublemma, thelattice isomorphism JfB± -^-JTB2 induces a correspondence between chambers whichpreserves dihedral angles. Since the dihedral angles determine the isometry type of aspherical simplex [GrBe] [theorem 5.1.2], there is a unique map of GAT(l)-spacesBx -> B2 which is isometric on chambers and induces the given combinatorial isomor-phism. Since the metric on each B4 is characterized as the largest metric for which thechamber inclusions are 1-Iipschitz maps, we conclude that our map Bx -> B2 is anisometry. In the general case, the buildings Bt may have a spherical de Rham factor Sjand split as B4 = S4 o B,'. The lattices JTB4 and 3fB\ are isomorphic: to a subcomplex C,'

of JfBl corresponds the subcomplex S4 o Q of Jf B4. The lattice isomorphismJ^~B[ = 3TBX -> Jf B2 = 3#~W2 is induced by a unique isometry B[ -> B'2 by the discussionabove. It follows that Dim B = Dim B'2 and Dim Si = Dim S2. Any isometry SL -> S2

gives rise to an isometry Bx -^B 2 which induces the isomorphism XBX ->JTB2. n

3.9. Geodesies, spheres, convex spherical subsets

We call a subset of a CAT(l)-space convex if with every pair of points with distanceless than TC it contains the minimal geodesic segment joining them. The followinggeneralizes corollary 3.5.2.

Proposition 3 . 9 . 1 . — Let C C B a convex subset which is isometric to a convex subset ofa unit sphere. Then C is contained in an apartment.

Proof. —- We proceed by induction on the dimension of B. The claim is trivialif dim(B) = 0. We assume therefore that dim(B) > 0 and that our claim holds forbuildings of smaller dimension than B.

Let A be an apartment so that the number of open faces in A which have non-emptyintersection with G is maximal. Suppose C $ A. Let p e G n A and q e C\A be pointswith pq $ Ej, A. Denote by V the union of all minimizing geodesies in A which connect pto its antipode p and intersect G — {p, p}; Vis a convex subset of A and canonicallyisometric to the suspension of Sj,(C n A) = Sp G n ~ZV A. By the induction assumption,there is an apartment A' through/* such that ~SV G £ Y,v A'. A' can be chosen to contain p.Then C n A s V s A ' and pq e 1tp A'. Hence the number of open sectors in A' inter-secting G is strictly bigger than the number of such sectors in A, a contradiction. There-fore G 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 E B B contained in an apartment. In particular

dim(K) < rank(B) - 1.

3.10. Convex sets and subbuildings

A subbuilding is a subset B' s B so that {i esf \ i(S) £ B' } forms an atlas for aspherical building structure; in particular B' is closed and convex.

Lemma 3 . 1 0 . 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 , Jt/') 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(j). If p e s then logj, maps I ink(s) isometrically to the join complement

ofJups in S p B(f) . Furthermore, ifpes lies in a W-face G of maximal possible dimension, then

there is a bijective correspondence between W'-chambers containing a, W-chambers ofB(s), chambers

of Link(.y), and ~Wp-chambers in "Zv B.

RIGIDITY OF Q.UASI-ISOMETRIES FOR SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS 145

Proof. — Let £ and % be interior points of faces in s with maximal dimension. ThenB(s) is the union of all geodesic segments of length TC from £ to %. Proposition 3.6.4implies that every pair of points in B(s) is contained in an apartment AC B(s).

Pick t0 e j / with s £ io(S), and set s/' ={ies/\ i|So = io |So}. Let W £ W bethe subgroup generated by reflections fixing s0 pointwise. According to lemma 3.1.1,the coordinate changes for the charts in stf' are restrictions of elements of W . Thereforej / ' is an atlas for a spherical building structure on B(s) modelled on (S, W ) .

Since .r0 £ 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(j) by section 3.3. Any twopoints in Link(j) lie in an apartment J S A C B ( J ) , SO logj, maps Link(^) isometricallyto the join complement of Hips in S^B^). The remaining statements follow. •

The building B(s) splits as a spherical join of the singular sphere s and a sphericalbuilding which we denote by Iink(j):

B(s) = soUnk(s).

Lemma 3 . 10 .2 . — If % e B and TJ lies in the apartment A £ B, then there is a % e Awith 7t = dfe, | ) = d(E,, rj) + d{t\, %). If d(^, vj) ^ n/2 then E, has an antipode in every top-dimensional hemisphere H C A.

Proof. — When Dim B = 0 the lemma is immediate. If d(£,, YJ) < -n then by induction

7j e Sn B has an antipode in £„ A. Therefore we may extend £•») to a geodesic segment £>]!

with 7)f C A of length 7t. The second statement follows by letting T) be the pole of thehemisphere. •

Proposition 3 . 1 0 . 3 . — Let C be a convex subset in the spherical building B. If C contains

an apartment then C is a subbuilding of full rank.

Proof. — By the lemma, any point Z, e C has an antipode % in G. By lemma 3.6.1,the union C^ g of all minimizing geodesies from £ to % which intersect G — { £, % } isa neighborhood of \ in C. In particular, for sufficiently small s > 0, C n BE(^) is a coneover S 5 C. Since f can be chosen to lie in an apartment Ao £ G by our assumption,and since the apartment S |A 0 in S | G corresponds to an apartment in C^ \, we seethat C is a union of apartments. It remains to check that any two points ^ i j e C liein an apartment contained in C. Choose an apartment A with t\ e A £ C. For T)£ G S^ G

there exists an antipodal direction in 2,, A and we can extend \t\ into A to a geodesic ?TQ|of length 7i. To the apartment S^A in £g C corresponds an apartment A' £ C^ |

containing £y]|. D3.11. Building morphisms

We call a map cp : B ->• B' between buildings of equal dimension a building morphismif it is isometric on chambers. Later, when looking at Euclidean buildings, we willencounter natural examples of building morphisms, namely the canonical maps fromthe Tits boundary to the spaces of directions.

19

A building morphism 9 has Lipschitz constant 1: 9 maps sufficiently short segmentsemanating from a point p isometrically to geodesic segments. Therefore it induces well-defined maps

(23) S p 9 : S , B - S 9 ( 1 ( ) B ' .

Since the chambers in B containing p correspond to the chambers in 1,p B (with respectto 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 9 spreading ifthere is an apartment Ao c B so that 9 Ao is an isometry.

Lemma 3 . 1 1 . 1 . — Let 9 : B - > B ' be a spreading building morphism. Then, 1 / ^ , | 2 e Bare points with <p(5i) = 9(^2) ='• £'> the images 0/" 2 5 9 and £ 5 a 9 in S5, B ' coincide.

Proof. — If 9 is spreading then each point £' e <p(B) has an antipode £' e (p(B).Any points ^eip"1^') and | e9~ 1 ( f ' ) are antipodes and minimizing geodesiesconnecting \ and % are mapped isometrically to geodesies connecting £' and f', i.e.9 B<5,t> • B(£, t) ->B'(S;', %') is the spherical suspension of the morphism £^ 9. Thereare canonical isometries persp^ ^ : S^ B -> Sg B and persp^, -g, : S5, B' -> SjT. B',cf. 3.6.1, and we have:

(24) 2 j 9 o persp5 -g = persp5,£, o 2 5 9.

The assertion follows. •

Lemma 3 . 1 1 . 2 . — Let 9 : B -> B' fo a spreading building morphism. Suppose ^ e B ,S; e B ' and set l[:= 9 ^ .

T^en fAere is an apartment A £ B containing £x such that 9 A is an isometry and theapartment A ' : = 9A £ B' contains £,%.

Proof. — Let us first assume that £2 e A^ = <pA2 where A2 is an apartment in B

such that 9 |Aj! is an isometry. Then there is a geodesic segment %,[ ^ ti of length 7t such

that ?2fi c A2 (lemma 3.10.2). Let %x eA2 be the lift of ti- By proposition 3.6.4, thesubbuilding B(^l512) contains an apartment A with 2 ^ A = S^ A2. The map 9 A isan isometry, because it is an isometry near \x. By construction, ^ e 9A.

The above argument implies that, since 9 is spreading by assumption, each point£i e B lies in an apartment Ax so that 9|A is an isometry. Therefore the assumptionin the beginning of the proof is always satisfied and the proof is complete. •

Corollary 3.11.3. — Let 9 be as in lemma 3.11.2. Then:

1. <p(B) is a subbuilding in B' .2. The induced morphisms S 5 9 are spreading.3. For all \x e B, ^ 6 9(B)> the™ ex^ts £2

6 9 " 1 £2 su°h that

(25) d^lx,l2) =dB.(<pZ1}Q-

4. If %2 satisfies (25) then there exists an apartment A e B containing \x, £2 such that 91A is an isometry.

Proof. — The first three assertions follow immediately from the lemma. We provethe fourth assertion:

By 1. we find a geodesic segment ?,[ E,!2 %[ of length n contained in <p(B). By 3. thereexists a lift %x of fi such that dB(^2, li) = dB,(^, %[). Applying the previous lemmato the morphism 2 5 9, which is spreading by 2., we find an apartment A s B(51; %x)

containing the geodesic segment \x £2 \x and so that S5 <p|s A, and therefore also 9 A,is an isometry. Q

Proposition 3 .11 .4 . — Let B and B' be spherical buildings modelled on Amod, and let

9 : B -> B' be a surjective morphism of spherical buildings so that 6B = 6B, o 9. Suppose T is

a face qfB and a' is a face ofB' contained in <p(B) so that 9T £ a'. Then there exists a face a

of B with x £ a and fa = a'.

Proof. — Let \ be an interior point of T and let ax be a face of B with 9 ^ = a'.Then ax contains (in its boundary) a point \x with 9 ^ = <p£, and by lemma 3.11.1there exists a face G containing \ (and therefore T) with 90 = ^al = a'. •

Corollary 3 .11 .5 . — Let B, B' W 9 be as in proposition 3.11 A. If h' C B' is a half-

apartment with wall m', and mC B lifts m', then there is a half-apartment K B containing m

which lifts K.

Proof. — Let T' C h' be a chamber with a panel cr' C m', and let csC m be the liftof a' in ?re. Applying proposition 3.11.4 we get a chamber T C B so that the half-apart-ment h spanned by T U m lifts K. D

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 Uo of a is defined as the subgroup of Aut(B, Amod) consisting

of all automorphisms g which fix every chamber C C B with the property that C <~\ a contains a

panel n <£ 8a.

We let GB 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.

l.If\Ja acts transitively on the apartments containing a for every root a contained in some apart-

ment Ao , then the group generated by these root groups acts transitively on pairs(G, A) where

C is a chamber in an apartment A £ B.

2. Suppose (B, Amcd) is irreducible and has dimension at least 1. Then the only root group element

g e Uo which fixes an apartment containing a is the identity.

Lemma 3 . 1 2 . 3 . — Let A and A' be apartments in the spherical building B. Then there

exist apartments Ao = A, A l 5 . . . , Ak = A' so that A j _ 1 n At is a half-apartment containing

A n A' for all i.

Proof. — Suppose that A and A' are apartments which do not satisfy the conclusionof 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 Arespectively A' strictly contains A n A'.

If A n A' is empty, we choose A" to be any apartment which has non-emptyintersection with both A and A'. If A n A' is contained in a singular sphere s ofdimension dim(A n A') < dim(B) we pick a chambers a C A and a' C A' withdim(<7 n s) = dim(<r' n s) = dim s. The subbuilding B(J) contains an apartment A"with s u a u a' C A" and A" has the desired property. It remains to consider the casethat A n A' contains chambers and is strictly contained in a half-apartment. Thenthere is a half-apartment h C A containing A n A' and so that I n A n A ' containsa panel 71. Let a' C A' be a chamber with a' n A n A' = TC. The convex hull A" ofh u a' is an apartment with the desired property. •

aProof of Proposition. — 1. Let GA be the group generated by the root groups U,

where a runs through all roots contained in an apartment A C B. If g e Uo thenGA = GBA because U,A = g^Jag~1 for all roots «C A. By lemma 3.12.3, given anyapartment A' there is a sequence Ao, ...,Ah = A' such that Aj_j n A j is a root.Hence GAo = GAi = . . . = GA, and it follows that GB = GA, for all apartments A'.

Let G± and a2 be chambers in B which share a panel TC = <JX n <r2. Since B is thick,there is a third chamber a with 5 n af = n. Pick apartments A^ containing a \J a{.Applying lemma 3.12.3 again, we see that there is a g e GB so that g(A1) = A2, andg fixes ff3. Hence gat = a2 and we conclude by induction that GB acts transitively onchambers.

Let Aly A2 be apartments and <T1( CT2 be chambers such that at £ Aj. By the aboveargument, there exists g e GB with gax = a2. By lemma 3.12.3 there is a g' e GB withg'igAj) = A2 and g' a2 = a2. Hence GB acts transitively on pairs C C A as claimed.

2. Since B is irreducible, there is a chamber a contained in the interior of a (seelemma 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 a bythe definition of Ua. Note that if iz and -K' are opposite panels in B', then B' containsevery chamber containing TT if and only if it contains every chamber containing TC'(lemma 3.6.1). Since for each panel TC there is a panel TCXC da in the same projectivityclass (see definition 3.6.2 and lemma 3.6.3) we see that B' contains every chamberin 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 a l l / 6 B' we have Sj, B' = Hp B,which implies that B' C B is open and consequently B' = B. •


Definition 3 .12 .4 . — A spherical building (B, A^^) is Moufang if for each root a C Bthe 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 o acts simply transitively on apartments containing a.

The spherical building associated with a reductive algebraic group ([Til, chapter 5])is Moufang. In particular, irreducible spherical buildings of dimension at least 2 areMoufang.

4. EUCLIDEAN BUILDINGS

There are many different ways to axiomatize Euclidean buildings. For us, thekey geometric ingredient is an assignment of Amod-directions to geodesies segments in aHadamard space. Just as with symmetric spaces, Amod-directions capture the anisotropyof the space, and they behave nicely with respect to geometric limiting operations suchas ultralimits, Tits boundaries, and spaces of directions.

4.1. Definition of Euclidean buildings

4.1.1. Euclidean Coxeter complexes

Let E be a finite-dimensional Euclidean space. Its Tits boundary is a round sphereand there is a canonical homomorphism

(26) p : Isom(E) -> Isom(3Tits E)

which assigns to each affine isometry its rotational part. We call a subgroupW ^ C Isom(E) an affine Weyl group if it is generated by reflections and if the reflectiongroup W : = p(Waff) C Isom(dTita E) is finite. The pair (E, Waff) is said to be a EuclideanCoxeter complex and

(27) an t .(E,W - r):=(an i .E,W)

is called its spherical Coxeter complex at infinity. Its anisotropy polyhedron is the sphericaldolyhedron

An oriented geodesic segment xy in a E determines a point in dTlts E and wecall its projection to Amod the /^^-direction of #y.

A wall is a hyperplane which occurs as the fixed point set of a reflection in W ^and singular subspaces are defined as intersections of walls. A half-space bounded by awall is called singular or a half-apartment. An intersection of half-apartments is a Weyl-polykedron. Weyl cones with tip at a point p are complete cones with tip at p for which theboundary at infinity is a single face in STite E.

Fix a point p e E. By W(/>), we denote the subgroup of W ^ which is generatedby reflections in the walls passing through^; W(^) embeds via p as a subgroup of W.

A Weyl sector with tip at p is a Weyl polyhedron for the Euclidean Goxeter complex(E, W (/>)); note that a Weyl sector need not be a Weyl cone, and a Weyl cone neednot be a Weyl sector. A subsector of a sector CT is a sector a' C a with dTita a' = dTits a;G lies in a finite tubular neighborhood of a'. A Weyl chamber is a Weyl polyhedron forwhich the boundary at infinity is a £^06. chamber; Weyl chambers are necessarily Weylcones. The Coxeter group W(/>) acts on 2^ E, so we have a Coxeter complex

with anisotropy map by

The faces in (Sp E, W(/>)) correspond to the Weyl sectors of E with tip at p.We call the Goxeter complex (E, W^) irreducible if and only if its anisotropy

polyhedron, or equivalently, its spherical Goxeter complex at infinity is irreducible.In this case, the action of W on the translation subgroup T < Waff forces T to be trivial,a lattice, or a dense subgroup. In the latter case we say that Wa(f is topologically transitive.

4.1.2. The Euclidean building axioms

Let (E, Waff) be a Euclidean Coxeter complex. A Euclidean building modelled on(E, Waff) is a Hadamard space X endowed with the structure described in the followingaxioms.

EB1. Directions. — To each nontrivial oriented segment xy C X is assigned a Amod-direction Q(xy) eAm o d . The difference in A.mod-directions of two segments emanating from thesame point is less than their comparison angle, i.e.

(28) d(Q(x^),Q(x-z))^2x(y,z).

Recall that given S1,82eAmod, D(S1; S2) is the finite set of possible distancesbetween points in the Weyl group orbits 6 ^ j . ^ ) and 0 ^ E ( ^ ) -

EB2. Angle rigidity. — The angle between two geodesic segments xy and xz lies in the finite

set D

We assume that there is given a collection s/ of isometric embeddings 1: E -»• Xwhich preserve Amod-directions and which is closed under precomposition with isometriesin W ^ . These isometric embeddings are called charts, their images apartments, and Ais 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 iA for an apartment A is well-defined up to pre-composition with an isometry a ep - 1 (W). Two charts iA , iA for apartments Aly A2

are said to be compatible if i^1 o iA is the restriction of an isometry in Waff. This holdsautomatically when Waff = p~1(W).

EBA. Compatibility of apartments. — The Euclidean coordinate charts for the apcrtmentsin X are compatible.

It will be a consequence of Corollary 4.6.2 below that the atlas s# is maximalamong collections of charts satisfying axioms EB3 and EB4.

We define walls, singular flats, half-apartments, Weyl cones, Weyl sectors, andWeyl polyhedra in the Euclidean building to be the images of the corresponding objectsin the Euclidean Coxeter complex under charts. The set of Weyl cones with tip at apoint x will be denoted hyiP~x. The rank of the Euclidean building X is defined to bethe dimension of its apartments. The building X is thick if each wall bounds at least3 half-apartments with disjoint interiors. We call X a Euclidean ruin if its underlyingset or the atlas J / is empty.

4.1.3. Some immediate consequences of the axiomsAxiom EB1 implies the following compatibility properties for the Amod-directions

of geodesic segments.

Lemma 4 . 1 . 1 . — Let x, y, z be points in X.

1. If y lies on x~z, then 8(xi) = Q(xy) = Q{yz).2. If xy, xz e SB X coincide, then 6(ry) = 6(#i).3. Asymptotic geodesic rays in X have the same Amoi-direction.

We call a segment, ray or geodesic in X regular if its Amod-direction is an interior

point of Amod.

Lemma 4 . 1 . 2 . — 1. If p e X and xi e X — p, then the ^ initially span a flat triangle

if Z.J)(x1, x2) > 0, and they initially coincide if / .„(#!, x2) = 0.2. If pi e X and ^ e 8Tlta X, then the rays p< £4 are asymptotic to the edges of a flat sector.

Proof. — 1. After extending the segments px^ to rays if necessary, we mayassume without loss of generality that xi e STita X. If z epxlt then Q(zxt) = Q(xt) so/.„(#!, #2) e D(0(*1), 6(#2)) which is a finite set. But /-t(x1} x2) -> Z.p(#i, x2) mono-tonically as z ->/>, which implies that Z-t(xu #2) = Lv{xx, x2), Z.t(p, x2) — TZ — /.„(*!, *2)when z is sufficiently close to p. Therefore A(/>, z, x2) is a flat triangle (with a vertexat 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

4.2.1. The Tits boundaryThe Tits boundary dTiu X is a GAT(l)-space, see 2 .3 .2 . Lemma 4 . 1 . 1 implies

that there is a well-defined Amod-direction map

(29) e,Tit jX:STltgX

which is 1-Iipschitz by (28).

Proposition 4 . 2 . 1 . — The space dTlte X carries a spherical building structure modelled on

the spherical Coxeter complex (#Tits E, W) with Amoi-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). If A is a Euclideanapartment in X then 8Tlts A is a standard sphere in dTjtB X. Clearly, any point £ e dTltB Xlies in a standard sphere. It remains to check that any two points \x and £2 in dTltB Xwith Tits distance TT are ideal endpoints of a geodesic in X. To see this, pick p e X andnote that the angle Z.,(5i, 52) increases monotonically as z moves along the ray p\x

towards \x. But by EB2 Z.g(?i, £2) assumes only finitely many values, so when z issufficiently far out we have Z-S(£i, £2) — Arits(£i» £2) — n> a n d the rays zZ* fit togetherto form a geodesic with ideal endpoints \x and £2. D

4.2.2. The space of directions

The space of directions Sa X is a CAT(l)-space (see section 2.1.3). Lemma 4.1.1implies that there is a well-defined 1-Iipschitz map from the space of germs of segmentsin a point x e X:

(30) eS;>x:S:X

In this section we check that this map induces a spherical building structure on SB X.By axiom EB2, 6 = 6S x satisfies the discreteness condition (17).

Lemma 4.2.2. — The space 2£X is complete, so 2£X = 2BX.

Proof. — Let (xk) be a sequence in X — { x } such that (xxk) is Cauchy in 2* X.Then Q(xxk) is Gauchy in Am0(1 and we denote its limit by 8. If Aj.C X is an apartmentcontaining xx~k then xxk e 2B Aj. C 2* X and 2,. Ak contains a spherical polyhedron ak

such that xxk e ffj. and Q\a/c; ak -»> Amod is an isometry. There is a unique \k e CTJ with^(^) — a n d we have d(Z,k, xxk) = dAmod(8, Q{xxk)) ->0. Hence (^) is Gauchy with0(£j.) = S and lim xxk = lim ^ in 2j.X. The discreteness condition (17) implies that(£j.) is eventually constant and therefore (xxk) has a limit in 2* X. •

We now apply proposition 3.5.1 to verify that 2,. X carries a natural structureas a spherical building modelled on (dTits E, W). The only condition which remainsto be checked is that antipodal points xxt and xx2 in 2X X lie in a subset isometric toS = dTita E. But l-x{xx, x2) = n implies that xxx2 = xxlux~x~2 and if AC X is anapartment containing xxx% then S e A C 2^ X is a spherical apartment containing xxx

and xx2.

Lemma 4 . 2 . 3 . — All standard spheres in S^X are of the form S a A where A is an

apartment in X passing through x.

Proof. — By corollary 3.9.2, standard spheres are A^-apartments, so we canfind antipodal regular points £1} £2

e <*• Then there is a segment xxxz through x with

3cxi = ^i. If A £ X is an apartment containing xx x2 then S ^ A n a 2 { ^ , ^ } andthe spherical apartments a and Tix A coincide because they share a pair of regularantipodes (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 s/th(x)and anisotropy map

(31) 6* :S.X->**<(*);

This structure is unique up to equivalence. The second reduction is analogous to thestructure constructed in proposition 3.6.4. We postpone discussion of this structureuntil 4.4.1 because we do not have an analog of lemma 3.1.1 in the case of nondiscreteEuclidean Goxeter complexes.

4.3. Product(-decomposition)s

Let Xj, i= 1, ...,n, be Euclidean buildings modelled on Coxeter complexes(Ej, W*ff) with atlases s/{ and anisotropy polyhedra &mo&. Then

acts canonically as a reflection group on E := Ex X . . . X En . We call the Goxetercomplex (E, W^) the product of the Coxeter complexes (E4, W^) and write

(32) (E, W^) = (E1; w y x . . . x (E., W^).

There are corresponding join decompositions

(33) (eW, E, W) = (5TltB E1 ; Wx) o . . . o (STlts En, W J

of the spherical Coxeter complex at infinity and

(34) Amod = A L d o . . . o A » o d

of the anisotropy polyhedron. The Hadamard space

(35) X = Xx X . . . X Xn

carries a natural Euclidean building structure modelled on (E, W^). The charts forits atlas s/ are the products i = ix X . . . X in of charts i4 e s/i. We call X equippedwith this building structure the Euclidean building product of the buildings Xj.

Proposition 4 . 3 . 1 . — Let Xbe a Euclidean building modelled on the Coxeter complex (E, Waff)with atlas s& 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 Euclidean Coxeter complexes so that a

segment ~xy C E is parallel to the factor E( if and only if its Amci-direction %{xy) lies in A'mod.

2. There is a decomposition (35) of X as a product of Euclidean buildings so that a segment

xyCE is parallel to the factor E4 if and only if its &mo6-direction 0(r>>) lies in ^!mod.

20


Proof. — 1. Proposition 3.3.1 implies that the spherical Goxeter complex at infinitydecomposes as a join

(36) (STlte E, W) = (Sl5 W J c . . . (S., W J

of spherical Coxeter complexes. By proposition 2.3.7, this decomposition is inducedby a metric product decomposition E = Ex X . . . X EB so that dTite E4 is cano-nically identified with Sj and, hence, a segment ~xj> C E is parallel to the factor E,if and only if Q(xy) 6 ^ , (36) implies that W ^ decomposes as the productW j- = W ^ X . . . X W"ff of reflection groups W^f acting on E4, thus establishingthe 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 ~xy C X is parallel to the factor Xj if and onlyif 6(xj)) e A ^ . Furthermore, the dTita X4 carry spherical building structures modelledon (fij,ltg E{, W4) so that the spherical building dTita X decomposes as the sphericalbuilding join of the dTita X4. Each chart i: E -> X, i e J / , decomposes as a productof A^j-direction preserving isometric embeddings \ : E4 -> Xj. The collection stfi ofall i, arising in this way forms an atlas for a Euclidean building structure on Xj 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 joindecomposition of the anisotropy polyhedron Amod into irreducible factors correspondsto unique minimal product decompositions of the Euclidean Coxeter complex (E, W^)and the Euclidean building X into irreducible factors. We call these decompositionsthe de Rham decompositions and the maximal Euclidean factors with trivial affine Weylgroup the Euclidean de Rham factors.

4.4. The local behavior of Weyl-cones

In this section we study the set Wv of Weyl cones with tip at p. The main result(corollary 4.4.3) is that in a sufficiently small neighborhood of/», a finite union of thesecones is isometric to the metric cone over the corresponding finite union of Amod facesin Sp X. This proposition plays an important role in section 6.

Let Wx and W2 be Weyl cones in X with tip at p. The Weyl cone W4 determinesa face 2 p W{ in the spherical building (Sj, X, Amod).

Sublemma 4 .4.1. — Suppose thai S , Wx = Sp W2 in Sp X. Then Wx n W2 is aneighborhood of p in W1 and W2.

Proof. —• According to lemma 4.1.2 each point in the face 2 p Wx = 2 p W2 isthe direction of a segment in Wx n W2 which starts at p. We can pick finitely manypoints in 2 p Wx = Sj, W2 whose convex hull is the whole face. The convex hull of thecorresponding segments is contained in the convex set Wx n W2 and is a neighborhoodof p in Wx and W2. •

Locally the intersection of Weyl cones with tip at a point p is given by theirinfinitesimal intersection in the space of directions 1ip X:

Lemma 4.4.2. — If W 1 ,W a e#" J ( , then there is a Weyl cone W sWv withW = Sp Wx n Sp W2. For every such W there is an e > 0 so that

W1 n W2 n B,(£) = W n B,(0.

fe intersection of Weyl cones with tip at the same point is locally a Weyl cone.

Proof. — By lemma 3.4.2 the intersection Sp Wx n S ^ W j is a Amod-face andhence there is a W fWv such that Ytv W = Sp Wx r> Sj, W2. By the previous sublemma,there are W4 e"#^, with W4 £ W4 and a positive e so that

w ; n Be(/>) = Wa n Be(/>) = W n Bt(p)

for any such W. If x is a point in Wj n W2 different from p then / ixeS, , W, soJ n W ; . Therefore

Wx n W2 n Be(/>) = WJ n Wa n Be(/>) = W n B,(^). n

Corollary 4.4.3. — i/" W l9 . . . , Wfc e ^ , fen fere w an s > 0 jacA ferf(U4 WJ n B2)(e) maps isometrically to {Hi Gp W4) n B(s) C G,, X via logp.

Proo/. — Let ^ denote the finite subcomplex of Sj, X determined by \J{ S,, W4.Pick ax, o2 6 ^ . By lemma 4.2.3 these lie in an apartment S,, Ao O2 c Sp X for someapartment ACTi O2 C X passing through p. If CTX is a face of Sp W4 and c2 is a face ofSj, Wj-, then by the sublemma above we may assume without loss of generality that(W?1 u W?2) n Bp(s) s AOiO2 where W?i (resp. W?2) is the subcone of W4 (resp. W,)with Sp W?1 = a1 (resp. S , W?a = a2). Since there are only finitely many such pairsGX, <r2 e ^ , for sufficiently small s > 0, every pair of segments px1}px2 £ U4 W4 boundsa flat triangle provided | px{ \ < s. n

4.4.1. Another building structure on 2 , X, and the local behaviorof Weyl sectors

Let a C Sp X be a Amod-apartment. By lemma 4.2.3 there is an apartment A C Xwith Sp X = a, 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 Wa C Isom(a).

Lemma 4 . 4 . 4 . — The reflection group W a contains the reflection group W^h coming from

the thick spherical building structure on Hp X.

Proof. — Let mC OL be a wall for the AJ^^) structure. There are apartmentsAjC X through p, i = 1, 2, 3, so that Sp Ax = a and the Sp At intersect in half-apart-ments with boundary wall m. By corollary 4.4.3 the pairwise intersections of the \are half-spaces near p. Choose charts iAi, iAa, iAj e si and let <p4i e W ^ be the unique

isometry inducing i^.1 o iA.. Then cp12 o <p23 o <p31 is a reflection at a wall w passing throughx = ij*{p) and satisfying Sj, iAl w = m. •

Fixing one apartment a C Sj, X, we take a chart i : S -> a from the atlas «s/th(p),and enlarge ^/^{p) by precomposing each chart i' e j/th(p) with elements of^ ' ( W J C Isom(S). Clearly this defines an atlas s/{p) for a spherical building structuremodelled on Amod(/>) = a/Wa.

Let A, At C X be apartments so that Sp A = a, Sp Ax = al5 and a n ax containsa chamber C C a. If iA, iA : E -»- X are charts from the atlas J / , then since A n Ax isa cone near /> by lemma 4.4.3, it follows that Sj,(iA o iA

x) : Sp A = a -> <xx = Sj, Ax

carries Wa faces in a to Wai faces in al5 while at the same time it carries Amod(p) facesof « to Amod(p) faces of ax. So every Amod(/>) face CTC S p X is a Wa, face for everyapartment a' containing a. Since the Wa,'s are all isomorphic, this clearly implies thatSj, W is a Amod(/>) face for every Weyl sector with tip at p. So we have shown:

Proposition 4 . 4 . 5 . — There is a spherical building structure (S p X, s&{p)) modelled on(S, Amod(/>)) so that Amod(p)-faces in S p X correspond bijectively to the spaces of directions ofWeyl sectors with tip at p. In particular, if A C X is any apartment passing through p, then thereis a 1-1 correspondence between walls mC A passing through p and Amo&(p)-walls in the apart-ment Sj, A, given by m i—> S p m. When X is a thick building, then s&(p) coincides with ^/^(p)

for every p 6 X.

Corollary 4 . 4 . 6 . — Corollary 4 . 4 . 3 holds when the Wt are Weyl sectors with tip at p.If Ax and A2 be apartments in X then A1 n A£ is either empty or a Weyl polyhedron. In particular,if A-y r\ A2 contains a complete regular geodesic then Ax = A2 .

Proof. — Each Weyl sector with tip at p is a finite union of Weyl cones with tipat p. Hence a finite union of Weyl sectors with tip at p is a finite union of Weyl coneswith tip at p, and the first statement follows.

If Ax, A2 C X are apartments and p e Ax n A2, then ~LV Q,x n T,p A2 is a convexAmod(/>) subcomplexof Sj, A4. Hence there are Amod(p) half apartments hlt .. .,hhCLv Ax

so that flj A4 = Sj, Ax n ~LV A2. By proposition 4.4.5, for each i there is a half-apartmentH, C A with Sp Hi = hi. Therefore Ax n A2 n Bp(s) = (f) H4) n Bp(s) and so Ax O A2

is a Weyl polyhedron near /». Consequently Ax n A2 is a Weyl polyhedron. •

4.5. Discrete Euclidean buildings

We call the Euclidean building X discrete if the affine Weyl group W ^ is discreteor, equivalently, if the collection of walls in the Euclidean Coxeter complex E is locallyfinite.

If p is a point in E then ap denotes the intersection of all closed half-apartmentscontaining/, i.e. the smallest Weyl polyhedron containing/. By corollary 4.4.6, eachaffine coordinate chart iA : E -> X maps ap to the minimal Weyl polyhedron in X which

contains iA(/>). Hence for any point x e X there is a minimal Weyl polyhedron ax

containing it. We say that x spans ax. ax is the intersection of all half-apartments containingx and, if X is thick, the intersection of all such apartments. The lattice of Weyl poly-hedra ay with x e ay is isomorphic to the polyhedral complex Jf"SB X.

Proposition 4 . 5 . 1 . — In a discrete Euclidean building X each point x has a neighbor-hood BS(A;) which is canonically isometric to the truncated Euclidean cone of height s over ~LX X.

Proof. —• Let iA : E -> X be a chart with x = iA(/>) and choose s > 0 so that anywall intersecting Bs(p) contains p. Then for any point y e Be(/>), the polyhedron ay

contains x and any apartment intersecting Be(/>) passes through x. Hence any twosegments xy and x~z of length < s lie in a common apartment and it follows that B£(/>)is isometric to a truncated cone. •

Assume now that Waff is discrete and cocompact. Then the walls partition Einto polysimplices which are fundamental domains for the action of Waff. This induceson X a structure as a polysimplicial complex. The polysimplices are spanned by theirinterior 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, thedimension of a flat is less or equal to the rank of X .

Proof. — Among the faces in dTits X which intersect the sphere 8Tita F we picka face a of maximal dimension. Then a n 8TltB F is open in 8TltB F. Let c be a geodesicin F with c(oo) e Int(ff) and let A be an apartment containing c. Then 3Tit8 A contains aand c{— oo) and convexity implies dTito F £ drVa A. Since F n A + 0, it follows thatF is contained in the apartment A. •

As a consequence, we obtain the following geometric characterization of apartmentsin 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 distancefrom an apartment approaches this apartment at a certain minimal rate given by theextent of its regularity.

Lemma 4 . 6 . 3 . — Suppose \ e dTUa X is regular and that the ray />£ remains at bounded

distance from an apartment F. Then every point x ep?, with

lies in F.


Proof. — Let y be a point on the ray nA(j>) %, and let z spy be the point where

the segment py enters A (we may have z =y). By lemma 4 . 1 . 2 Z 1 ( j » , A ) > 0 , and by

lemma 3 .4 .1 we have /-z{p, A) ^ dAmod(Q(pz), 8 Amod). The comparison triangle A (a, b, c)

in the Euclidean plane for the triangle A(/>, •rcA(/>)> z) satisfies Z.b(a, c) ^ TC/2 and

t-Afi, b) > d^jflijz), 8 AmJ._Hence d(p, A) ^ d(p, z) Sm(dAmJQ(pz), 8 Amod)). SinceQ(pz) = 0(gji>) -> 0(/>£) asjc epZ, tends to oo, the claim follows. •

Corollary 4 . 6 . 4 . — Each complete regular geodesic which lies in a tubular neighborhoodof an apartment A must be contained in A. IfAx and A2 are apartments in X and A2 lies in a tubularneighborhood of Ax , then Ax = A a .

Another implication of the previous lemma is the following analogue of lemma 4 . 4 . 2at infinity.

Lemma 4 . 6 . 5 . — If Glt C 2C X are Weyl chambers with dTits Cx = 8Tlts G2, thenthere is a chamber G £ Gx n G2.

Proof. — It is enough to consider the case that the building X is irreducible. Theclaim is trivial if the affine Weyl group is finite and we can hence assume that Waf

is cocompact. If p is a regular geodesic ray in Gx then, by the previous lemma, it enters G2

in some point p and Cx n G2 contains the metric cone K centered at/> with ideal boundarydTita K = dTltB C4. Since W ^ is cocompact, K clearly contains a Weyl chamber, n

Proposition 4 . 6 . 6 . — There is a bijective correspondence between apartments in X and8TlUj X given by

Proof. — We have to show that every apartment K in 8Tita X is the boundaryof a unique apartment in X. Since K contains a pair of regular antipodal points, thereis a regular geodesic c whose ideal endpoints lie in K. c is contained in an apartment A.Since the apartments dTlts A and K have antipodal regular points in common, theycoincide 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 e A, it can

be extended into A.

Proof. — If 7] is the direction of c at p then, by lemma 3.10.2, 7) has an antipodein the spherical apartment Sp A. Hence c has an extension into A. Q

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 8Tlte A. In particular, it is contained in a finite union of apartments passing

through x.

Sublemma 4 . 6 . 9 . — Let Y be a Euclidean building with associated admissible sphericalpolyhedron Amod. Then for each direction § e int(Amod) the subset Q~1(8) in the geometricboundary dK Y is totally disconnected with respect to the cone topology.

Proof. — Suppose that y,y',y" G Y so that Q{yJ') = Q(JY') = 8 . Define thepoint z by yy' r\yy" =yz. If z =¥y',y" then the angle rigidity axiom EB2 implies that/-z(y',y") > a0 : = 2c'dAmod(8, d Amod) and by triangle comparison we obtain:

Sin a0

As a consequence, for each z e Y the closed subset { £ e dx Y | 6(£) = 8 and z eyZ,} of0~ *(8) is also open and we see that each point in 6-1(8) has a neighborhood basis consistingof open and closed sets. •

4.7. Subbuildings

A subbuilding X' c X is by definition a metric subspace which admits a Euclideanbuilding structure. This implies that X' is closed and convex and that dTitg X' is aspherical subbuilding of dTitg X which is closed with respect to the cone topology. Weconsider a partial converse:

Proposition 4 . 7 . 1 . —• Let X be a Euclidean building and B s dTite X a subbuilding of

full rank. Then the union X ' of all apartments A with 8Tlta A s B has the following properties:

• If X ' is closed then it is a subbuilding of full rank and the subbuilding dTita X ' s 8Tlts Xis the closure B o / B with respect to the cone topology. Furthermore, X ' is the unique subbuildingwith dTiu> X ' = B.

• IfX. is discrete or locally compact then X ' is closed.

Proof. — Observe that

X' u { A apartment | 0Tlto A s B } = u { A apartment | 8TIte A £ B }.

We first show that X' is a convex subset. Consider points x1} x2 e X ' . There are apart-ments Aj with x1 e A4 s X'. By lemma 3.10.2, there exist ^ e 9Tltg A4 withAe.-(*3-i> ?i) = ^ The canonical map 41 : ^uta -^ -»• S^ X is a building morphismand satisfies the assumption of proposition 3.11.2. Thus, since Z-^^i, £2) = n> thereis an apartment dTUs A £ X' which contains 5i> ?a and projects isometrically to S^ Xvia <\i. This means that A;X e A. Consequently xx x% C A and X' is convex. Similarly, oneshows that any ray and geodesic in X' lies in an apartment A which is limit of apart-ments An with dTlts An s B, i.e. dTlts A s B and A s X'. The building axioms areinherited 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 apartmentA £ X', and if x' is sufficiently close to x then A contains x. Hence X' is closed in this case.


Assume now that X is locally compact and that (xj C X' is Cauchy with limitx 6 X. Let p e X' be some base point. Any segment p~x~n lies in some apartment An £ X'and we can pick rays px'n \n in An so that lim x'n = x and 6£n = Qpx. After passing toa subsequence, we may assume that (£ J converges to a point \ e B. Since 6|B = 6£,lemma 4.1.2 implies that the segments p\n n p\ C X' n p\ converge to p\. Hence p\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 finiteHausdorff distance from F then F and F' bound a flat strip, i.e. an isometrically embeddedsubset of the form F X I with a compact interval I C R. In this case, the flats F and F'are called parallel. Consider the union P r of all flats parallel to F; Pp is a closed convexsubset of X and splits isometrically as

Proposition 4 . 8 . 1 . —- The set PF is a subbuilding of X and Y admits a Euclidean buildingstructure.

Proof. — By proposition 4 .6 .1 , PF is the union of all apartments which contain Fin a tubular neighborhood, and 8Tltg Pp is the union of all apartments in STit8 X whichcontain the sphere dTits F. The subset 9Tito PF s dTits X is convex by lemma 4.1.2 anda subbuilding by proposition 3.10.3. Proposition 4.7.1 implies that PF is a subbuil-ding of X. As a consequence, the Hadamard space Y inherits a Euclidean buildingstructure. •

If dim(F) = rank(X) — 1, then Y is a building of rank one, i.e. a metric tree.Since Sv Y is in this case a zero-dimensional spherical building, any two raysji% and J ^in Y either initially coincide or their union is a geodesic. This implies:

Lemma 4 . 8 . 2 . — (i) Let H1 and H 2 be two flat half-spaces of dimension rank(X) whoseintersection H x O H 2 coincides with their boundary flats. Then Hx u H 2 is an apartment.

(ii) If A1} A2 , A3 c X are apartments, and for each i + j the intersection A^ n A ; isa half-apartment, then A± n A2 n A3 is a wall in X.

Lemma 4 . 8 . 3 . — Let G l 9 C2 , G3C 8Tlta X be distinct adjacent chambers, withTZ = C1 r> C2 n G3 their common panel. Then there is a p e X so that if

C o n e ( / > , i t ) = U { ^ | | l e u ) ,

then logp^Cj) C Sp , X are distinct chambers for every p' e Cone(/», 7t) and any apartment A C X

such that STite A contains two of the C{ must intersect Cone(/>, TC).

Proof. — Let m C 9Tita X be a wall containing the panel TC. Then each chamber Qlies in a unique half-apartment h{ bounded by m, and pairs of these half-apartments


form apartments. Let Aij be the apartment in X with dTits Aw = A, u h} • Bylemma 4.8.2, H Aij is a wall M C X, and we clearly have dTitB M = m. If p e M,then the half-apartments logj, Af C Sp X are bounded by logj, m = Tiv M, so they aredistinct; otherwise fl Ai} 4= M. Hence the chambers logp G4 C logp ht are distinctchambers.

If A C X is an apartment with G4 u C3 C 3Tits A, i =(= j , then there are chambersC ^ C j C A n Ai3. with 9Tit8 C = C o 8Tita G3- = C3-. The Tits boundary of the Weyl poly-hedron P = Atj n A contains Ct u C3-, so it intersects Gone(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, Waff), with atlas s/'. The affine Weyl group may be reduced to a reflection subgroup

W^j C Waff if there is a Waff compatible subset s#' C J / forming an atlas for a Euclidean

building modelled on (E, W ^ ) .

In contrast to the spherical building case, the affine Weyl group of a Euclideanbuilding does not necessarily have a canonical reduction with respect to which itbecomes thick. For example, a metric tree with variable edge lengths does not admita thick Euclidean building structure. However, there is always a canonical minimalreduction, 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 thereis a unique minimal reduction W^f C W ^ so that (X, E, W ^ ) splits as a product H X{ whereeach Xj 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 Weylgroup has a fixed point) or their affine Weyl group is cocompact.

Proof. — We first treat the case when (0Tits X, Amod) is a thick irreducible sphericalbuilding of dimension at least 1.

Step 1. — Each apartment A C X has a canonical affine Weyl group GA . If A C X isan apartment, a wall M C A is strongly singular if there is an apartment A' C X so thatA o A' is a half apartment bounded by M. Since dTite X is thick and irreducible, forevery wall m C 3Tlts A there is a strongly singular wall M C A with dTits M = m.

Sublemma 4 . 9 . 3 . — The collection J?A 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 ~Zp M C Sp Xis a wall with respect to the thick building structure (Sp X, A |^ (/>)); this is becauseany half-apartment h C Sp X with boundary S p M can be lifted to a half-apartment

21


H C X with boundary M, Sj, H = h by applying proposition 3.11.4 to the surjectivespherical building morphism logp : 2Tlte X -> Sp X.

If M l 5 M2C A are strongly singular walls intersecting at p e A, then ~Sp Mt is a^Wd(^) w a ^ ^n 2 , A C Sj, X, and so if we reflect Sp M2 in Sp Mx (inside the apart-ment Sj, A), we get another A^od(/») wall which is then the space of directions of thedesired strongly singular wall M3.

Now suppose that M l 5 M2 eJ?A are parallel. Amod is irreducible so there is astrongly singular wall M3 intersecting both M4 at an acute angle. Reflect M2 in M3

to get M4, reflect M3 in Mx to get M5, and M4 in M.1 to get M6, and finally reflect M6

in M6 to get a wall which is the image of M2 under reflection in M.1. The walls M4 areall in ^#A, so we are done. •

Proof of Proposition 4.9.2 continued. — Hence for every apartment AC X thecollection of strongly singular walls in A gives us a group GA C Isom(A) which is gene-rated by reflections.

Step 2. — The group GA is independent of A. Since 8Tlia GA C Isom(^Tlte A) is anirreducible Goxeter group, it follows that GA is either a discrete group of isometriesor it has a dense orbit. When GA is discrete, it is generated by the reflections in thestrongly singular walls which intersect a given GA-chamber in codimension 1 faces.When GA has a dense orbit, it is generated by all the reflections in strongly singularwalls passing through any open set. If two apartments Ax and A2 intersect in an openset, it follows that GAi is isomorphic to GAa; therefore GA is independent of A. So thereis a well-defined Goxeter complex (E, W ff) attached to X.

Step 3. — Finding (E, W ff) apartment charts. If Z is a convex domain in an apart-ment A C X and i : 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 i to an isometry of a convex set Z C E onto Z.

Pick an apartment Ao C X and an isometry i0 : E -> Ao which carries W^C Isom(E)

to GA Q . Then restrict to a W^f chamber C0C E and its image Go = io(Co) C Ao. Givenany chamber G C X, there is an apartment Ax containing subchambers of G and Go.There is a unique isometry ix : E -> At so that i ~x and ijf1 agree on the subchambersCo n A0C A l5 and a unique isometry i c : E D G -> G so that i^1 and if1 agree on thesubchamber C n Ax. If A2 is another apartment with dTits Go, 9Tits C C 9Tlt8 A2, we getanother isometry i2 :E->-A2 ; but the convex set Ax n A2 contains subchambers ofCo and G, so i]~1 and i% J agree on a subchamber of G. Therefore ic is independent ofthe choice of apartment asymptotic to Co u G.

Sublemma 4 . 9 . 4 . — Let A C X be an apartment, and let C l 5 G2C dTIte A be adjacent

Amoi-chambers (G1 n G2 is a panel). For i = 1, 2 we let ic (A) : E -> A be the unique

isometric extension of 15. where C4 C A is a W'^-chamber with dTlta C4 = G4. Then

Proof. — For i = 1, 2 let A . C X be an apartment with Go u Q C dTlta Aj. IfCx is contained in the convex hull of Co u C2 (or G2C ConvexHull(C0 u Cx)) thenCx u G2C dTite(A n A2), so the sublemma follows from the fact that i^.1(A) restrictedto A n A2 coincides with I2"

1|A/-IA • So we may assume that there is a chamberC"3 C dTlta Ax n dTita A2 which meets Gx and G2 in the panel n = Cx n C2. Bylemma 4.8.3 (applied to the original Euclidean building (X, E, W^)) , there is apointy € Ax n A2 so that Cone(p, TC) C AX n A2 and logJ)(Ci) C ~Sp X are distinct chambersfor i = 1, 2, 3. Therefore if1 and lif1 agree on Cone(j!>, u). Hence the isometries ^ ( A ) , loi

1

agree on Cone(/», TT), which means that i^1(A) o ic (A) : E ->E is a reflection. Butsince Sp(Cone(/>, 7c)) = log^C^ n log,(G2) n logp(G8), Cone(/>, TT) spans a stronglysingular wall in A and so the reflection iT 1(A) o ic (A) e W ^ . D

Proof of Proposition 4.9.2 continued. — By sublemma 4.9.4, we see that for eachapartment A C X, there is a canonical collection of isometries i : E -> A which aremutally W ^ compatible, and which are compatible with the ic : C -> G for everychamber G C A. We refer to such isometries as W^-charts, and to the collection ofW^-charts (for all apartments) as the (E, W^) atlas sf'.

Sublemma 4 . 9 . 5 . — Let A l 5 A 2 C X be apartments with d-dimensional intersectionP = Ax n A2 . If p e P is an interior point of the Weyl polyhedron P, then there is an apartmentA 3 C X so that A3 contains a neighborhood of p e P, and A3 n A{ contains a Weyl chamber.

Proof. — We have Sp A1 n 2 p A2 = 2 , P by lemma 4.4.3. Let ax C 2 p P be ad — 1-dimensional face of 2 P P, and let <r2 be the opposite face in ~ZV P. If TXC S P AX

is a chamber containing G1} then we may find an opposite chamber T 2 C SJ, A2. Butthen T2 contains a face opposite aly and this must be a2 since each face in an apartmenthas a unique opposite face in that apartment. Let C4 C e>Tlte A, be the chamber suchthat log,, G4 = T4 . Then there is a unique apartment A3 C X with C [ U C 2 C 9Tlts A3.2j,PC Sj, A3, so A3 has the properties claimed. D

Proof of Proposition 4.9.2 continued. — If Ax, A2 C X are apartments with A t n A2 # 0,then any W ^ charts i4: E -> Aj are W ^ compatible since by sublemma 4.9.5 wehave a third apartment A3 C X so that i2 and i2 are both W ^ compatible with i3 : E -> A3

on an open set U C Ax n A2. Hence s/' gives X the structure of a Euclidean buildingmodelled on (E, W^). From the construction of W^f it is clear that (X, J / ' ) is thick.

Step 4. — The case when X is a l-dimensional Euclidean building, i.e. a metric tree. LetA0C X be an apartment, dTita Ao = { v)i, 7j2}. For each p e X let TCAO(/>) e AO be thenearest point in Ao, and pAg e Ao be a point (there are at most two) with^(/'AO* "AOC/O)

= d(p, Ao). Let Jl C Ao be the set of points pkfj where p e X is a branchpoint: | 2j, X | ^ 3; let GC Isom(A0) be the group generated by reflections at pointsi n ^ . For each % e 8Tlia X y/h there is a unique isometry i from the apartment Ao = T)X YJ2to the apartment 7)x \ which is the identity on the half-apartment YJ1 t\% n ^ %. If \x 4= 2>


then we have two isometries ilt i2 : Ao -*• £2 where i, * agrees with t5. on T]J £, n £x £2.By inspection ij- 'oi , e G. Hence for each apartment A C X we have a well-definedset of isometries Ao ->• A. As in step 3 it follows that these isometries are G-compatible,so they define an atlas J / ' for a Euclidean building structure on X.

Step 5. — X is an arbitrary Euclidean building modelled on (E, Waff). Let W = 8TitB W ^ ,

and let W ' C W C Isom(gTlt8 E) be the canonical reduced Weyl group of 8Tlbl X givenby section 3.7. Let WaffC Isom(E) be the inverse image of W under the canonicalhomomorphism Isom(E) -> Isom(^Tita E). Let 6' : 8Tiis X -> A ^ = S/W' be theA^j-anisotropy map. We may define A^-directions for rays xE, C X by the formula0'(x£) = 6'(5) eA^^ . We define the A^-direction of a geodesic segment icjiCX bysetting f)'(xy) = 6'(x^) for any ray x\x extending xy; if x\% is another ray extending Icythen \x e 3Tits X and £2 e dTlte X are both antipodes of t\ e 8Tlta X where _yq is a rayextending yx, so 6'(xy) is well-defined. The remaining Euclidean building axioms followeasily from the fact that any two segments px, ~py initially lie in an apartment A C X(corollary 4.4.3) and for our compatible (E, W^) apartment charts we may take allisometric embeddings i: E -*• X for which dTita i: 8Tita E -> dTlta X is an apartmentchart for (aTiteX,A;od). _

We may now apply proposition 4.3.1 to see that (X, E, W^) splits as a productof Euclidean buildings (X, E, W,,,) = (IIX,, I lE,, I IW^) so that each 8Tlta X, isirreducible. Let ( W ^ ' C W ^ , J / { be the canonical subgroup and atlas constructedin steps 1-4, and set W;ff = II (W^) ' C Isom(E), j / ' = I I j / , . Then (X,, E(, (W^)', s/Jhas the properties claimed in the proposition. Fix an apartment Ao C X and a chartiAo e j / . If Ao, . . . , Afc = Ao is a sequence of apartments so that Aj_, nAj is a half-apartment for each i, then there is a unique isometry gf: A,_1 -> Aj so that gt is theidentity on A4 _ x n A j . Axiom EB4 implies that gi o ... o gt o iAo e stf for each i, so inparticular ^ = ^ 0 . . . ogx etAo,(Waff). From the construction of (Wj^)' it is clearthat the group of all such isometries g : Ao -> Ao contains ^ ( W ^ ) C Isom(A0) whereiAo 6 J / ' . So W^ffC W ^ is a minimal reduction of W^. •

4.10. Euclidean buildings with Moufang boundary

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 0 C Aut(B, A ,^ ) and every g e Uo there is a unique automorphism

£ x : X -H» X JO that 8Tlis gx = g. In other words, if G is the group generated by the root

groups, then the action of G on 8Tits 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 e Uo is nontrivial. IfA s X is an apartment such that #Tita AD a, then g{A) n Ais a half-apartment; moreover Fix(^) n A = £(A) n A.

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 Euclideanbuilding of rank > 2. Therefore A,^ is a spherical simplex with diameter < n/2 andthe faces of dTlts X define a simplicial complex.

Lemma 4 . 1 0 . 2 . — Let A C X be an apartment, p0 e X, p e A the nearest point in A,and a C 8A a root. Then the stabilizer of p0 in the root group XJa fixes p .

Proof. — Using lemma 3.10.2 extend the geodesic segment pop to a geodesicra.yp0 E, = pop u/>£ so that the ray p\ lies in the half apartment Cone(/>, a) C A. If g e Ua

fixes p0) then it fixes the ray p0 \, and hence the half-apartment Cone(/>, a). DWe now assume that the spherical building (9Titg X, Amod) is Moufang. Pick

p e X, and let (Sj, X, Aôd(/»)) denote the thick spherical building defined by the spaceof directions ~LP X with its reduced Weyl group (see section 3.7). Suppose H + C Xis a half-apartment whose boundary wall passes through p, h+ = Sj, H + C 2 , X is a&mod(P) root, and let a+ = dTita H + C BTits X. If Uo is the root group associatedto a+, and Vo C Uo is the subgroup fixing p, then we have a homomorphism

Lemma 4 . 1 0 . 3 . — The image of Vo+ is the root group Uft+ associated with h+, and thisacts transitively on apartments in S p X containing h+. In particular, (Lp X , Aôd(/>)) is a thickMoufang spherical building.

Proof. — By corollary 3.11.5, if h_ C 2 p X is a AÔfr) root with

Bh_ = dh+ = S,(0H+),

then there is a half-apartment H_ C X so that H_ and H + have the same boundaryand 2,, H_ = A_. Given two such Amod(/>) roots M., M C 2,, X so that #_ u h+ formsan apartment in Sp X, we get two half apartments H*_ so that H'_ u H + forms an apart-ment in X. Since (e>TItg X, Amod) is Moufang, the root group UA+ C Aut(3Tlts X, A ^ )contains an element which carries HL to H i . By 3.12.2, g " extends " uniquely to anisometry g : X -> X which carries the apartment HL u H + to the apartment H i u H + ,fixing H + (see 4.10.1). It remains only to show that the isometry S p g : Y,p X ->• S s Xis contained in the root group Ufc+C Aut(S1) X, A*^(/»)). Clearly X^g fixes h+. LetC C S p X be a Aôd(/>) chamber such that C n h+ contains a panel TT with it cj: 8h+.Using proposition 3.11.4 we may lift C to a (subcomplex) C C dTlts X so that G n da+

maps isometrically to G n SA+ under logj,: fiTlt9 X -> Sp X. ^ fixes an interior pointof C, so Sp g fixes an interior point of C, which implies that T,p g fixes C as desired. •

Definition 4.10.4. — A point s e X is a spot if either1. The affine Weyl group W ^ has a dense orbit or2. W ^ is discrete and s corresponds to a 0-simplex in the complex associated with X .

If A c X, then Spot (A) is the set of spots in A.


Lemma 4 . 1 0 . 5 . — If A C X is an apartment, p0 e A is a spot, then for every p #= p0 thereis a root a C 8Tlts A and age U o so that g fixes p0 but not p .

Proof. — For each A^od(/>0) root k+ C 2 ^ X we have a singular half-apartmentH + C A with EJ)o H + = h+, and this gives us a root a+ = STits H + C 8TiiB X, the rootgroup Uo+, and the subgroup Vo+C Uo+ fixing p0. By lemma 4.10.3, the image of Vo+

in Aut(2£0X,A*d(A,)) is the root group UA+. Since (SM X, A£od(/>0)) is Moufang*the group GPo generated by the Vft+'s as h+ runs over all A^od(/»0) roots in 1,p A actstransitively on A ^ , ^ ) chambers in SPo X (see 3.12.2). If ^ e X — p0 is fixed byevery Vo+, then pop e HVo X is fixed by G^, which means that it lies in every A*od(/)0)chamber of 2 ^ X, forcing pop e S^ A. Hence the point q eA nearest p is differentfrom p0, so we may find a singular half-apartment H + C A containing p0 but not q(because p0 is a spot), and use the root group U8x;t ^ to move q while fixing H + . Thiscontradicts the assumption that p is fixed by every Vo . •

Proposition 4 .10 .6 . — Let X be a thick, nonflat Euclidean building of rank at least two,and suppose dTiUs X is an irreducible Moufang spherical building. Let G C Aut(#Tits X, Amod)be the subgroup generated by the root groups of dTitB X, and consider the isometric action « / G m 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 e X lies in the apartment A C X if and only if p is the unique spot in X which isfixed by the stabilizer of p in Uo for every root a C dTitB A.

3. If A C X is an apartment, and a C 8Tite A is a root, then as g runs through all non-trivialelements of\Ja, we obtain all singular half-apartments H C A with dTlis H = a as subsetsA n Fixfe).

Proof. — Let M s G be a maximal bounded subgroup. By the Bruhat-Tits fixedpoint theorem [BT], M has a nonempty fixed-point set Fix(M), which contains a spotsince when W ^ is discrete the fixed point set of a group of building automorphismsis a subcomplex. By lemma 4.10.5, we see that if p0 e Fix(M), then maximality of Mforces Fix(M) = {p0 }. Conversely, if p0 e X is a spot, then the stabilizer of p0 has fixedpoint set {p0} by lemma 4.10.5, and by the Bruhat-Tits fixed point theorem, thestabilizer is a maximal bounded subgroup.

For every p e X and every apartment A C X, let G(p, A) be the group generatedby the stabilizers of p in the root groups Ua, where aC 8TiU A is a root. If p e A C Xis a spot, then by lemma 4.10.5 we have Fix(G(/>, A)) = {p}. If p $ AC X, then thenearest point/>0eA top is contained in Fix(G(/>, A)) by lemma 4.10.2; hence Fix(G(^, A))contains a spot other than p0.

Claim 3 follows from property 2 of proposition 4.10.1, the fact that #TlteX isMoufang, and the fact that every singular half-apartment is the intersection of twoapartments, in


Definition 4.10.7. — -(/"AC X is an apartment, then the half-apartment topology onSpot (A) is the topology generated by open singular half-apartments contained in A.

With the half-apartment topology, Spot(A) is discrete when W ^ is discrete andcoincides with the metric topology when Waff has dense orbits.

5. ASYMPTOTIC CONES OF SYMMETRIC SPACESAND EUCLIDEAN BUILDINGS

In this section we arrive at the heart of the geometric part in the proof of ourmain results. We show that asymptotic cones of symmetric spaces and ultralimits ofsequences of Euclidean buildings (of bounded rank) are Euclidean buildings.

Our main motivation for choosing the Euclidean building axiomatisation EB1-4is that these axioms behave well with respect to ultralimits. Indeed, the Euclideanbuilding axioms EB1, EB3 and EB4 which are also satisfied by symmetric spaces, i.e. theexistence of Amod-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 onlytechnical 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 sphericalbuildings. Intuitively speaking, the rescaling process involved in forming ultralimitspulls the spherical building structure (the missing angle rigidity property) from infinityto the spaces of directions.

5.1. Ultralimits of Euclidean buildings are Euclidean buildings

Theorem 5 . 1 . 1 . — Let X B , s e N , be Euclidean buildings with the same aniso-tropy polyhedron Amod. Then, for any sequence of basepoints * n 6 X n , the ultralimit(X u , *Q) = co-lim(Xn, •„) admits a Euclidean building structure with anisotropy polyhedron

Proof. —• The building XM is a Hadamard space (lemma 2.4.4). A Euclideanbuilding structure on XM consists of an assignment of Amod-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 Rham factor. Thegeneral case allowing a Euclidean de Rham factor is a trivial consequence.

EB1. — We can assign a Amod-direction to an oriented geodesic segment in Xu

as follows. A segment x^y^ arises as ultralimit of a sequence of segments xnyn in X,and we define the direction as:

(37) 6 ( ^ X ) : = co-Hm 6(*^T)

The ultralimit (37) exists because Amoa is compact. Inequality (28) in EB1 passes tothe ultralimit:

. *J-

This implies that the left-hand side of (37) is well-defined and

Thus axiom EB1 holds. It implies lemma 4 .1 .1 . Therefore, segments which containa given segment have the same Amod-direction and we can assign Amod-directions togeodesic rays.

EB2. — Since geodesies are extendible in X u , it suffices to show:

Lemma 5.1.2. — If xaeXa and S,a,r}ae 8Tits Xm then LxJ$,a, TJJ is contained in

Proof. — The rays xa £u and xu 7)w are ultralimits of sequences of rays xn £n

and xn 7]n in XB and we can choose £„, t\n e dTlta Xn so that 6(5J = 6(xu £u) and®(?ln) = 6(^m ^u). Let pn : [0, oo) -> Xn be a unit speed parametrisation for the geodesicray xn \n. The angle Z_p (()(?B, •/)„) is non-decreasing and continuous from the right in t(lemma 2.1.5) and, since XB satisfies EB2, takes values in the finite set D. For rfeDset / B ( r f ) : = m i n { ^ 0 : Z P n ( ( ) ( ^ , > ) J > r f } 6 [ 0 ) c o ] and ta{d) : = «-lim tn(d). Thenthere exist doeD and T > 0 with ta{d0) = 0 and 2T ^ ta{d) for all d> d0. Thepoints x'n := pn(tn(d0)) and xB' : = pB(T) satisfy for w-all n, x'a : = w-lim x'n = xu,A; ' : = to-lim A;B + ^w and the ideal triangle A(#B, xn', 7jB) has angle sum n. By a versionof the Triangle Filling Lemma 2.1.4 for ideal triangles in Hadamard spaces, A(#J,, x'l, Y)B)can be filled in by a semi-infinite flat strip SB. The ultralimit to-lim SB is a semi-infiniteflat strip filling in the ideal triangle A(#u, x'^, 7jM) and therefore

as desired.

— After enlarging the affine Weyl groups of the model Goxetercomplexes of the buildings Xn , we may assume that the Xn are modelled on the sameEuclidean Coxeter complex (E, Waff) whose affine Weyl group Wa(f contains the fulltranslation subgroup of Isom(E), i.e. p~1(W) = W ^ where p : Isom(E) -> Isom(5Tlta E)is the canonical homomorphism (26) associating to an afline isometry its rotationalpart. (Here we use that the Xn do not have Euclidean factors.)

The atlases J / B for the building structures on XB give rise to an atlas for a buildingstructure on Xu as follows: If iB e s/n are charts for apartments in XB so thatco-lim d(in(e), *n) < oo for (one and hence) each point e e E , then the ultralimitiw : = «-lim iB: E -> XM is an isometric embedding which parametrises a flatin XM. The collection «s/M of all such embeddings iu satisfies axiom EB3 in viewof lemma 2.4.4. Axiom EB4 holds trivially, because coordinate changes i"1 o betweencharts itt, i e s#a are Amod-direction-preserving isometries between convex subsetsof E and such isometries are induced by isometries in p-1(W) = Waff. Hence s/a is

an atlas for a Euclidean building structure on X u 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, Waff)and denote by W ^ 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 Rham factor. Then

any asymptotic cone X u inherits a Euclidean building structure modelled on (E, Waff). The

building Xm is thick if X is thick and the affirm Weyl group W ^ is cocompact.

Proof. — We have Xm = <o-lim(XB, * J where the XB are scale factors withto-lim XB = 0, XB is the rescaled building XB XB and *„ e Xn are base points; Xw inheritsthe Euclidean building structure modelled on (E, W^) which was constructed in theproof of the previous theorem.

Suppose now in addition that X is thick and W ^ is cocompact. Then any wallwnC XB branches, i.e. there are half-apartments HBiC XB, i — 1, 2, 3, so that the inter-section of any two of them equals wn and the union of any two of them is an apartment(lemma 4.8.2). If a sequence of walls wn satisfies «-lim d(wn, *B) < oo, it follows thatthe ultralimit of the sequence (wn) is a branching wall in Xw . Since W ^ is cocompactby assumption, there is a positive number d so that any flat in X, whose ideal boundaryis a wall in dTltB X, lies within distance at most d from a branching wall in X. In viewof 6>-lim \ = 0, this implies that any flat in Xw , whose ideal boundary is a wall indTits X u , is a branching wall. Thus, the Euclidean building structure on XM is thick. •

5.2. Asymptotic cones of symmetric spaces are Euclidean buildings

We start by recalling some well-known facts from the geometry of symmetricspaces 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 Hadamardmanifold, i.e. a complete simply-connected Riemannian manifold of nonpositivesectional 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 andacts transitively on X. A k-flat in X is a totally geodesic submanifold isometric toEuclidean A-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 therank of X. We will call the maximal flats also apartments. Pick an apartment E in X andlet W ^ be the quotient of the set-wise stabiliser StabG(E) by the pointwise stabiliserFixG(E). Then W ^ can be identified with a subgroup |of Isom(E). This subgroup isgenerated by reflections at hyperplanes and contains the full translation group. Wecall (E, W^) the Euclidean Goxeter complex associated to X. Its isomorphism typedoes not depend on the choice of E, because G acts transitively on apartments. Considerthe collection of all isometric embeddings i: E -> X so that W ^ is identified withStabG(i(E))/NormG(i(E)). Walls, singular flats, Weyl chambers et cetera are defined

22

as images of corresponding objects in E via the maps i. Note that the singularflats are precisely the intersections of apartments. The induced isometric embeddings^Titsl: Tits E -*• dTitg X form an atlas for a thick spherical building structure on8TlteX modelled on the spherical Goxeter complex (aTlte E, W) = dTlts(E, Wafr). Thegroup W is isomorphic to the Weyl group of the symmetric space X. Composing theanisotropy map 68TitsX : dTite X -+ Amod with the map SX -> dTiia X which assigns to everyunit vector v the ideal endpoint of the geodesic ray 11-» exp(to) one obtains a natu-ral map

(38) e

from the unit sphere bundle of X to the anisotropy polyhedron A ,^ . We will call Q(v)the An^-direction of o e S X ; Amod-directions of oriented segments, rays and geodesiesare defined as the A^^-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 ofpoints. Let Sj, X be the unit sphere at p e X, equipped with the angular metric, andlet Gp be the isotropy group of p. Then 8 induces a canonical isometry SPIGp ~ Amod

where SPIGp is equipped with the orbital distance metric. The quotient map Sp X -*• Amod

is 1-lipschitz and, for any x,y e X we have the following counterpart to inequality (28):

(39)

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 EuclideanCoxeter complex (E, W ^ ) . Then, for any sequence of base points *n e X and scale factors XB

with to-lim Xn = 0, the asymptotic cone Xm = w-lim(Xn X, *„) is a thick Euclidean buildingmodelled on (E, Waff). Moreover, Xw is homogeneous, i.e. has transitive isometry group.

Proof. — EBl. — Let Amod be the anisotropy polyhedron for (E, W^). Theconstruction of Amod-directions for segments in X u is the same as in the building case.We define directions by (37) and (39) implies that the definition is good and that EBlholds.

EB3 and EB4. — The Euclidean Goxeter complex (E, W^) is invariant underrescaling, because W^C Isom(E) contains all translations. Apartments in X u andtheir charts arise as ultralimits of sequences of apartments and charts in X, and axioms EB3and EB4 follow as in the building case, cf. section 5.1.

EB2. — The only nontrivial task is to verify the angle rigidity axiom EB2. Thiswill be done in the following lemma.

Lemma5.2.2. — If p eXu andxu x2<=Xa- {p}, then Lv(xx, xt) e

Proof. — If z'hep~x\—p and z'k ->p, then Z ^ f o , xz) -+ Lv{xx, x2) and

L%-k{p, x2) -> TC — Lv{xXi xz) by lemma 2.1.5. Since 0(2^ x2) -> 0 ( ^ ) we can find x'lk e z'kxu

4 * 6 4 *2, and p'k e z ^ such that Lt<k{x[k, x'2k) -> Z.p{x1} x2), /L,k{p'k, x'2k) -> n - Lv{xx, x2),

and 8 ( 4 ^ ) = Q(z'k x2) ->6(/Sx^). Since geodesic segments in Xffl are ultralimits ofgeodesic segments in XB X, we can find sequences pk, xlk, x^, zk e X such that zk epkxlk,

and finally |2 t*u.| , | zk x^ |, | Zj./»fc | -»- oo. Applying a sequence of elementsgjc^G = (Isom(X))<> we may assume in addition that zk is a constant sequence, zk = o.Hence the sequences of segments OA;1S., OX^, "op~k subconverge to rays o£l5 o£2, and ovjrespectively, which satisfy the following properties:

2. Z^ltB(^i, y < Z . ^ , *2) , Z.Tits(iri, ^2) < 7c — / .„(*!, xa) by lemma 2 . 3 . 1 ;

3. o^! u o>i is a geodesic, so ZTi ts(^1, 7)) = TC.

We conclude that

Ap(xu xt) = Aihfo, y £D(6(y3 6(5,)) = D(6(M), 6(^))

as desired. DHence we have constructed a Euclidean building structure on X u . Since G acts

transitively on Weyl chambers in X, it follows that the isometry group of Xw actstransitively on Weyl chambers in X u ; in particular, X o is homogeneous. To see thatthe building structure on Xw is thick it is therefore enough to check that the inducedspherical building structure of S.u X u modelled on (dTlts E, W) is thick. One wayto see this is to construct a canonical isometric embedding a of the thick sphericalbuilding 8Tlts X modelled on (dTlts E, W) into S,M X M by assigning to S; e dTita Xthe initial direction in * u of the geodesic ray w-lim *„ i; in Xm. That a is isometricfollows, for instance, from the definition (8) of the Tits distance. This finishes the proofof the theorem. •

6. THE TOPOLOGY OF EUCLIDEAN BUILDINGS

In this section, X will denote a rank r Euclidean building. The main goal of thesection is to understand homeomorphisms of X. As motivation for the approach takenhere, consider a closed interval I topologically embedded in an R-tree T. Because everyinterior point p e I — dl of the interval disconnects T, every path c : [0, 1] -> T joiningthe endpoints of I must pass through//, i.e. c([0, 1]) 2 I. A similar phenomenon occursin X if we consider topological embeddings of closed balls B C X of dimension equalto rank(X): if [c] e H r(X, 8B) and [8c] e H^^dB) is the fundamental class of 8B, thenthe image of the chain c contains B. By using 4.6.8, we can construct such c so thatImage(c) — U is contained in finitely many flats, where U is any given neighborhoodof SB. It follows that any b e B — 8B has a neighborhood V6 in X such that B n V6

is contained in finitely many flats.


6.1. Straightening simplices

If Z is a Hadamard space, there is a natural way to " straighten " singularsimplices a : Afc -> Z (cf. [Thu]). Using the usual ordering on the vertices of the standardsimplex, we define the straightened simplex Str(a) by " coning ": if Str(a|A _ ) hasbeen defined, then Str(a) is fixed by the requirement that on each segment joiningp e Ai._1 with the vertex opposite AA_1 in Aj., Str(a) restricts to a constant speed geodesic.The simplex Str(a) lies in the convex hull of the vertices of a. This straightening operationinduces a chain equivalence on C,(Z). By using the geodesic homotopy between Str(or)and a, one constructs a chain homotopy H from the chain map Str to the identity withthe property that Image(H(o)) £ GonvexHull(Image(o)) for any singular simplex a.

When Z is the Euclidean building X, then it follows from lemma 4.6.8 that forevery singular chain c e Cj.(Gone(X)), Image(Str(c)) is contained in finitely manyapartments.

Corollary 6 . 1 . 1 . — i/" V £ U £ X are open sets, then Hfc(U, V) = 0 for everyk> r = rank(X).

Proof. — If [c] e Hj.(U, V), then after barycentrically subdividing if necessary,we may assume that the convex hull of every singular simplex in c (respectively dc) liesin U (respectively V). The straightened chain Str(c) determines the same relative classas c since Image(H(c)) C U, Image(H(dc)) C V and

Str(c) —e = 8H(c) + H(ftr).

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. D

Lemma 6 . 1 . 2 . — Let Z be a regular topological space, and assume that Hj.(U l5 U2) = 0for every pair of open subsets U 2 £ U x £ Z, k > r. If Y £ Z is a closed neighborhood retract

and U C Z is open, then the homomorphism H r(Y, Y n U ) - > H f (Z, U) induced by the

inclusion is a monomorphism. In particular, the inclusion Y -> Z induces a monomorphism

H r(Y, Y — y) -> H r (Z, Z —y) of local homology groups for every y s Y.

Proof. — If [cj e Hf(Y, Y n U), then there is a compact pair (Kl5 K2) s ( Y , Y n U)and [fj eH r (Ki , K2) so that »,([cj) = [cj where i: (Kl5 K2) -* (Y, Y n U) is theinclusion. If [cj is in the kernel of Hr(Y, Y n U ) - > Hr(Z, U) then there is a compactpair (K l5 K2) = (K3, K4) £ (Z, U) such that ./,([*J) = 0, wherej : (Kl5 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 W l5 W2 C Z such that Y — U £ W l5 K4 £ W2; this is possiblesince Y — U is closed, K4 is compact, and Z is regular. Shrink V if necessary so that

r - i (Y _ U) C Wx. We now have: Hr(Y, Y n U ) ^ Hf(V, r ' f Y n U J J i s a monomor-


phism since r is a retraction; Hr(V, r~1(Y n U)) -> Hr(V u W2, r~1(Y n U) u W8) is anisomorphism by excision; Hr(V u W2, ^ ( Y n U) u W2) ->H r(Z, r - ^ Y n U) u W2)is a monomorphism by the exact sequence of the triple (Z, V u W2, r~1(Y nXJ) U W2)and H r + 1(Z, V u W8) = 0. It follows that [cj = 0. n

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 Hj.(Z, Y), then we define Support(Z, Y, [c]) C Z — Y to be theset of points z e Z — Y such that the image of [c] in the local homology groupHj.(Z, Z — { z }) is nonzero. Support(Z, Y, \c\) is a closed subset in Z — Y, and containedin the image of the chain c.

Lemma 6 . 2 . 1 . — Let B be a topologically embedded closed r-ball in X, Y a subset containingSB, and denote by \x the image of a generator ofHr (B, SB) induced by the inclusion (B, SB) -> (X, Y).Then Support(X, Y, y.) = B - Y.

Proof. — We may apply lemma 6.1.2 since B is a closed (absolute) neighborhoodretract. Therefore Support(X, Y, ji) coincides with Support(B, B n Y , [ B ] ) = B - Ywhere [B] denotes the generator of Hr(B, SB) which is mapped to y,. O

Now let U be an open subset of X and consider [c] eH r (X, U). After subdividingthe chain c if necessary, we may assume that the convex hull of each simplex of 8c iscontained in U, so that [Str(c)] = \c\. By 6.1, cx = Str(c) is carried by a finite unionof apartments 0*, so [c] is the image of [cj] e H r (^ , n U ) under the inclusionH f ( ^ , ^ n U ) ->H f(X, U). Applying lemma 6.1.2 to the neighborhood retract 0>,we find that the inclusion Support^, ^ n U , [cj]) in X coincides with Support(X, U, [c]).Hence we have reduced the problem of understanding Support(X, U, [c]) to a problemabout supports in the finite polyhedron SP.

Recall that ~LP X has a thick spherical building structure with anisotropy poly-hedron A^rtiP) (see section 4.2.2).

Lemma 6 . 2 . 2 . — Pick pe0\\J. When s > 0 is sufficiently small, logp mapsSupport^, &> n U, [cj) n B^s) isometrically to (u, C(Q)) n B(s) C G(S,, X) = G,, X,where the C4C Hp X are A^od(/>) chambers and G(G{) C Cp X is the cone over G4.

Proof. — The set 0* is a finite union of apartments, so by corollary 4.4.3 whens > 0 is sufficiently small logj, maps & n Bj,(s) isometrically to (u4 Cp At) n B(e) C Cj, X,where the A4 C are the apartments passing through />. We may assume thatUC X\Bp(s). Then [cj determines a class 02] e H r ( ^ n B!p(s), n 8Bp(s)). Theunion u4 Sp A{ C Sj, X has a polyhedral structure induced by the thick buildingatlas £?tu{p), and this induces a polyhedral structure on the pair (01 n BJej, 0> n SBp(e)).

The r-dimensional faces of this polyhedron are (truncated) cones over Aj£od(/>) chambersin the A*od(/>) subcomplex u{ S , A4 C ~LV X. Hence the lemma follows from elementaryhomology theory. D

Corollary 6 . 2 . 3 . — If B is a topologically embedded r-ball in X, then for every p e X\SBthere are finitely many Aôd(/>) chambers Cj C ~LV X so that logp maps B n Bp(s) isometricallyto (u4 G(G4)) n B(s) C C j X /or sufficiently small e > 0.

/Voo/l — Let (A e H r(B, SB) be the relative fundamental class. Lemma 6 .2 .1implies that Support(X, SB, [[x]) = B\SB and the corollary follows from lemma 6 .2 .2 . •

6.3. The topological characterization of the link

If Z is a topological space and z e Z , then we say that two subsets Sj, S2C Z havethe same germ at z if Sx n N = S2 n N for some neighborhood N of z. The equivalenceclasses 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 SP-îx)of germs of topological embeddings of Rf passing through x e X. Let SP2{x) be thelattice of germs generated by SP-^x) under finite intersection and union.

Lemma 6 . 3 . 1 . — The lattice SP%{x) is naturally isomorphic to the lattice J^S,. X generated

by the Aôd(jc) faces of S e X under finite intersection and union.

Proof. — By lemma 6.2.2 we know that elements of SP-^x) correspond to finiteunions of A*od(x) chambers in SB X. Intersections of A*od(x) chambers yield A*od(x)faces of S8 X, so we have a well-defined map of lattices 3 : <S 2(

X) -*" "^x -^ DY takingeach element of • ^ W to its space of directions at x (which is a finite union of A*od(x)faces). S is injective by Corollary 4.4.3. The image of S contains the apartments inJfZx X, and since (Za X, jtftb) is a thick spherical building every A*od(x) face of SB Xis an intersection of apartments, and hence S is onto. •

6.4. Rigidity of homeomorphisms

In this section we prove the following results about homeomorphisms of Euclideanbuildings:

Proposition 6 . 4 . 1 . — A homeomorphism of Euclidean buildings carries apartments toapartments.

Note that homeomorphic Euclidean buildings must have the same rank since therank 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 9 : Y = X X E" -> Y' = X ' X En ' a homeomorphism. Then n = n',

and 9 carries fibers of the projection Y ->• X to fibers of the projection Y' -*• X ' inducing a hotneo-morphism 9 : X -> X' .

Theorem 6 . 4 .3 . — Let X = I l f ^ X ^ , X ' = n | = 1 X ; be thick Euclidean buildingswith topologically transitive affine Weyl groups, and irreducible factors X i 5 X^. Then a homeo-morphism 9 : X -»• X ' preserves the product structure.

Theorem 6 .4 .4 . — Let X, X ' be irreducible thick Euclidean buildings with topologicallytransitive affine Weyl group, and suppose rank(X) ^ 2. Then any homeomorphism X -> X ' wa homothety.

6 .4 .1 . The induced action on links

Let X, X' be Euclidean buildings, and let 9 : X -*• X' be a homeomorphism.Pick a point x in X, and set x' = <p(x) e X'. The homeomorphism 9 induces an iso-morphism of lattices SP2(x) -> 2(

x') (see section 6.3) and therefore a dimension-preserving isomorphism Jf<px : -^S^. X -> Jf2B, X' of lattices. By proposition 3.8.1the lattice isomorphism Jffx is induced by an isometry 2 a 9 : 2B X -> 2X, X'.

6.4.2. Preservation of flats

Consider a singular &-flat F. Its germ at a point x e F is a subcomplex of 3#~%x X.The image of this subcomplex L under Jfya is the subcomplex L' associated to thegerm of 9(F) in J^"L^X) X; L determines a standard (&— 1)-sphere in Sx X. Since• ^ j , . is induced by an isometry Y>x 9 : SB X -> S,,, , X, L' determines a standard(& — 1)-sphere in S¥(ai) X. This sphere is the space of directions of a singular &-flat F'.9(F) and F' coincide locally, because their germs coincide. Hence 9(F) is a completesimply-connected metric space which is locally isometric to Euclidean &-space E*.Therefore, 9(F) is isometric to E*.

6.4.3. Homeomorphisms preserve the product structure

Let X, X' be Euclidean buildings which decompose as products

x=nxi) x'=nx;

of thick irreducible Euclidean buildings Xj, X.'f with almost transitive affine Weylgroup. We have a corresponding decomposition of the spherical buildings £„ X andS,,, X' into joins of irreducible spherical buildings:

We recall that this metric join decomposition is unique, cf. proposition 3.3.3, andtherefore for each x e X the isometry 2 , 9 : Sa X ->• S?{x) X ' decomposes as a join

Za 9 = o 2,. <p, of isometries Se % : 2^. Xj -» 2(ip((1!)) Xg(i) where a is a permutationof { 1, . . . , k }. In particular, X and X' have the same number of irreducible factors.We claim that the permutation a is independent of the point x. To see this, note thatany two points y, z e X lie in an apartment A and consider the map between apart-ments 9|A : A -> 9(A) (compare section 6.4.2). A parallel family of singular flats in Ais carried by 9|A to a continuous family of singular flats in 9(A); since there are onlyfinitely many parallel families of singular subspaces, we conclude by continuity that9|A carries parallel singular flats to parallel singular flats. Consequently the permu-tation <y is independent of x as claimed. We assume without loss of generality that a isthe identity. Our discussion implies that a singular flat contained in a fiber of the pro-jection /»j: X ~> Xj is carried by 9 to a flat in a fiber of the projection p[ : X' -> X,'.Therefore each fiber of the projection pt: X ->• X4 is carried by 9 to a fiber of theprojection p[ : X' -»• XJ. Hence for each i there is a homeomorphism 94: X4 -> X,'such that 94 o pi = p\ o 9, and it follows that 9 = ITi tp4.

6.4.4. Homeomorphisms are homothetiesin the irreducible higher rank case

Let X, X' be as in theorem 6.4.4. Let A be an apartment in X and consider thefoliations of A by parallel singular hyperplanes. Since X is irreducible of rank r, wecan pick out r + 1 of these foliations J^o, ..., J^r such that the corresponding collectionof 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 toirreducibility.

The image of A under 9 is an apartment A' and the foliations ^ are carriedto foliations J^' of A' by parallel singular hyperplanes. Note that these are also r-inde-pendent, since any /•-fold intersection of mutually non-parallel hyperplanes belongingto these foliations is a point. Choose affine coordinates x1} ..., xr for A such that theleaves of <3f0 are level sets of xx + . . . + xr and the leaves of the foliation J^ for t > 1are level sets of xi. Choose similar coordinates x[, ..., x'r on the target A' so that9({ #4 = 0 }) = { x[ = 0 } and 9({ 1ixt = 1 }) = { Z*f = 1 }. Consider those leaves in Awhich contain lattice points. Since 9 maps leaves to leaves one sees by taking successiveintersections 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 andhence, by continuity, an R-linear isomorphism.

We now know that 9|A : A -*• A' is an affine map preserving singular subspaces.Angles between singular subspaces are preserved, because the isomorphisms of simplicialcomplexes 3^^z are induced by isometries. Hence the simplices { xi ^ 0, S ^ ^ 1 } and{ x[ ^ 0, SA:,' < 1 } are homothetic and 9 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 9 is a homothety.

6.4.5. The case of Euclidean de Rham factors

We now consider Hadamard spaces X = Y X E" where Y is a thick Euclideanbuilding of rank r — n with almost transitive affine Weyl group. Clearly lemma 4.6.7continues to hold for X, and so do lemma 4.6.8 and the homological statements insection 6.1. Applying the reasoning from section 6.2 we conclude:

Lemma 6 . 4 . 5 . — Every topologically embedded r-ball in X is locally a finite union

U4 G4 X E" where the Q C Y are Weyl chambers.

It follows that every closed subset of X which is homeomorphic to E" is a unionof de Rham fibers, since its intersection with each fiber of p : X -» Y is open and closedin this fiber. If x e X, we may characterize the fiber of p : X -*• Y passing through xas the intersection of all closed subsets homeomorphic to E" which contain x.

Now let X' = Y' X E"', where Y' is a thick building of rank r' - n'. If X'is a homeomorphism, then we have r = r' by comparing local homology groups. Sincethe fibers of the projection maps p : X -> Y, p' : X' -> Y' are characterized topologicallyas above, we conclude that Y' of quotient spaces.

7. OUASIFLATS IN SYMMETRIC SPACESAND EUCLIDEAN BUILDINGS

In this section, X will be a Hadamard space which is a finite product of symmetricspaces and Euclidean buildings. We have a unique decomposition

(40) X = E " x I I X ii

where n e No and the X4 are non-flat irreducible symmetric spaces or Euclidean buildings.The maximal Euclidean factor E" is called the Euclidean de Rham factor. An apartmentis by definition a maximal flat and splits as a product of apartments in the factors. Allapartments in X have equal dimension and it is called the rank of X. Singular flats aredefined as products of singular flats in the factors. If the building factors are thick, thensingular flats can be characterized as finite intersections of apartments. Note that theonly singular flat in E" is E" itself and hence every singular flat in X is a union of deRham fibers.

7.1. Asymptotic apartments are close to apartments

Proposition 7 . 1 . 1 . — Let 2. be a family of subsets in X with the property that for any

sequence of sets Q,n e i , base points qn e Qn, and scale factors dn with co-lim dn = 00, the

ultralimit co-limB(i~1 -Q_n, qn) is an apartment in the asymptotic cone c o - l i m ^ ^ - X , qn). Then

there is a positive constant D so that any set Q, e St is a D Hausdorff approximation of a maximal

flat F(QJ in X.23

Proof. — Let us consider a single set Q, in J and choose a base point <? e Q,. Theultralimit co-Km(»~1-Q/, q) is an apartment in the asymptotic cone co-lim (n^-X, q)which contains the base point * : = (q).

Step 1. —- We first show that Q, is, in a sense to be made precise, quasi-convexin regular directions. Let xaya be a regular segment in <o-lim(«~1-Q>, q) which contains* as interior point. xaya is the ultralimit of a sequence of segments xnyn in X withendpoints xn,yn e Q_. There is a compact set A C Int(Amod) which contains the directionsof co-all segments xnyn. Let Fn be a maximal flat containing the segment xnyn. (FB isunique for co-all n.) Pick s > 0 so that d(A, d Amod) > s. Denote by Dn the diamond-shaped subset of all points p e FM so that Z.Xn{p,y^) < s and ^-Vn(p, xn) < s.

Sublemma 7 . 1 . 2 . —• There exists r> 0 so that for u>-all n the sets D n are contained in

the tubular r-neighborhood of Q_.

Proof. — We prove this by contradiction: Choose a point zn e Dn at maximaldistance dn from Q, and assume that co-lim dn = oo. Then the asymptotic coneco-lim^"1-X, zn) = Gone(X) contains the apartments F' : = co-limn d~1-¥n andF" : = co-limn d~x-Q. The point za = (zn) is contained in F' but not in F" and thereforeF' and F" are distinct apartments in Cone(X). Let za x'a (respectively zay'a) be theultralimits of the sequences of segments zn xn (respectively znyn). By the choice of thepoints zn, the points x'a and j ^ are contained in F" u ^ F". Since we can extendincoming geodesic segments in apartments according to 4.6.7, we may assume withoutloss of generality that x'^^y'^ e dx F". Let Wx and W2 be the Weyl chambers in Cone(X)centered at za which are spanned by the rays rx : = za x'a and r2 : = zay'a • By thechoice of e and the definition of Dn , the rays rx and r2 yield in the space of directionsS^ Cone(X) interior points of antipodal chambers. Consequently, the union W2 u W2

contains a regular geodesic c passing through za. Since dx W< n dK F" contains theregular point ^(oo), the chamber dx W4 is entirely contained in 8X F". Thus the idealendpoints c(± oo) of c are contained in dx F" and we conclude by 4.6.4 that cC F"and hence za e F", a contradiction. D

Step 2. — Suppose y n e Q , and to-lim n~1-d(q, qn) = 0.

Sublemma 7 .1 .3. — We have <o-lim d(qn, D J < oo.

Proof. — The constant sequence q and the sequence qn yield the same point inthe ultralimit w-lim (n -^X, q), which is an interior point of co-lim {rrl-~Dn,q).Therefore

If <o-lim d(qn, D J = oo, then

F : = <o-lim (d(qn, D . ) " 1 ^ . , qn) s co-lim (d(qn, D J - ^ F . , qn)

is a complete apartment in co-lim (d(qn, D J ^ - X , qn) (by (41)) which lies at unitdistance from w-lim qn e (d(qn, D J - ^ Q , qn). But the latter is also an apartment inw-lim (d(qn, D J ^ - X , qn) and we obtain a contradiction to Corollary 4.6.4. •

We now know that there is an rx > 0 such that for every R > 0, Q, n B4(R) C Nr,(Dn)for to-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 r2 such that for every R,Q, n Ba(R) and Dn n Ba(R) are r2-Hausdorff close to one another for w-all n.

Sublemma 7 . 1 . 4 . — For every R > 0, D n n Ba(R) form an u>-Cauchy sequence (*) with

respect to the Hausdorff metric.

Proof. — Suppose X is a symmetric space. Since for w-all n the sets Dn O Ba(R)have mutual Hausdorff distance < 2r2, if the sublemma were false we could find Haus-dorff convergent subsequences of { Dn } with distinct limits. The limits would be distinctmaximal 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 sequenceskn, ln -> oo and a radius R so that the Hausdorff distance between D^ n Ba(R) andD, n Ba(R) remains bounded away from zero. Then the w-lim(D4ij, q) and co-lim(D,n, q)are distinct apartments in the Euclidean building w-lim(X, q) lying at finite Hausdorffdistance from one another, contradicting corollary 4.6.4. •

By the sublemma, w-lim Dn n Ba(R) exists for all R (as an w-limit of a sequencein the metric space of subsets of Ba(R) endowed with the Hausdorff metric) and so weobtain a maximal flat F C X with Hausdorff distance < r2 from Q_.

Step 4. — We saw that each set Q,in 2. is the Hausdorff approximation of a maximalflat F(Q_). Denote by d(Q) the Hausdorff distance of Q, and F(QJ. Assume that thereis a sequence of sets Q_n e 2. with lim d(Qn) = oo. Choose base points un e X so that un iscontained in one of the sets Q n or F(Q,J but not in the tubular ^(Q.J/2-neighbourhoodof the other. Then the apartments w-lim rf(Q,n)~

1-Qin and w-lim rf(Q_J~1-F(Q_J havefinite non-zero Hausdorff distance in the asymptotic cone o)-lim(<i(Q_n)~

1-X, un). Thiscontradicts 4.6.4. The proof of the proposition is now complete. •

Corollary 7 . 1 . 5 . — There is a positive constant Do = D0(L, C, X, X') such that for

any (L, C)-quasi-isometry X ' and any apartment A in X, the image <p(A) is a D0-Haus-

dorff approximation of an apartment A' in X ' .

(x) A sequence xn in a metric space X is co-Cauchy if a subsequence with full co-measure is Cauchy. If X iscomplete, then we define co-lim xn to be the limit of this subsequence.

Proof. — According to proposition 6.4.1, for any sequence of basepoints and anysequence of scale factors \ , the asymptotic cone (D of O carries apartments to apart-ments. We can apply proposition 7.1.1 to the collection J of all images 9(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 Euclideanbuilding of rank r, with model polyhedron Amod, and Y will be an arbitrary Euclideanbuilding with model polyhedron Amod.

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 Q_ in X is contained in a -tubular neighbourhood of a finite union of maximal flats,Q.C N p ( U F e ^ F ) where card(#") < p.

Corollary 7.2.2.-—- The limit set of an (L, C) r-quasiflat Q, in X consists of finitely manyWeyl chambers in dTita X; the number of chambers can be bounded by L and G.

Lemma 7 .2 .3. — Let P C Y be a closed subset homeomorphic to Rr. Thus P is locallyconical (by corollary 6.2.3,), so it has a well-defined space of directions "Lp P for every p e P.We have:

1. If p e P then every v e Sp Y has an antipode in Sj, P.

2. If w e Sj, P, then there is a ray pt,C P, £ e 8TltB Y such that pi = w.

Proof. —• Since P is locally a cone over a ~Lp P, we have H r_1(S3)P) ~ Z, andthe inclusion Sj, P ->• Sp Y induces a monomorphism H f_1(SJ)P) -> H r_1(SJ) Y) sinceTtp Y is an r — 1-dimensional spherical building. Now if the first claim were not true,then 1ip'PC ~LpY would lie inside the contractible open ball Bv(tz) CSpY, makingH^S.PJ-H^^Y) trivial.

The second claim now follows from the first by a continuity argument: w is thedirection of a geodesic segment contained in P since P is locally conical, and a maximalextension of this segment must be a ray. •

Although we will not need the following corollary, we include it because its proofis 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 r then P is contained in a finite numberof apartments. The number of apartments is bounded by the bilipschitz constant of P.

Proof. — Let a e Amod be the barycenter of Amod, and consider the collectionof rays with Amod-direction a contained in P. Since P is bilipschitz to Er, a packingargument bounds the number of equivalence classes of such rays (we know that theTits distance between distinct classes of rays is bounded away from zero (cf. 4.1.2)).

Let if C 8TiUt Y be the (finite) set of Weyl chambers determined by this set of rays,and let ^"be the finite collection of flats in Y which are determined by pairs of antipodalWeyl chambers in S?. We claim that P is contained in UF e r F. To see this, note thatif p e P then by lemma 7.2.3 we can find a geodesic contained in P with Amod-direction awhich starts at p. This geodesic has ideal boundary points in =9", so by 4.6.3 the geodesiclies in U F e r F. D

Another consequence of lemma 7.2.3 is

Corollary 7 .2.5. — Pick a e Amod and L, C, s > 0. Then there is a positive real number Dsuch that if Q_C X is an (L, G) r-quasiflat, y e Q_, and R > D, then there is z e Q, with

Proof. — If not, then there is a sequence Qk of quasiflats, yk e Qk, and Kk -> oosuch that for every zk e Qk with | d(yk, zk) — Rs | < sRj. we have Z.(0(jvft Zj.), a) ^ s.Taking the ultralimit of R^"1 • Qk C R^* • X we get y^ e Qa C XB and for every 2W 6 Q_M

with | d(ya, za) — 1 | < e we have Z.(0(jyw za), a.) s. But this contradicts lemma 7.2.3since Q,w is bilipschitz to Er: we can pick v e SVu Q,m with 0(o) = a and find a geodesic

segment jym ^m C Q_M with ya za = v, and for w — all k zk satisfies the conditions ofthe lemma. •

Lemma 7.2.3 implies that quasi-flats " spread out": a pair of pointsy0, z0 lyingin a quasi-flat Q,C X can be extended to an almost collinear quadruple j l 5 jy 0 , z0, zx

while maintaining the regularity of Amod-directions. To deduce this we first prove aprecise statement for Euclidean buildings.

Lemma 7 . 2 .6 . — Let <xx e Amod be a regular point, and let sx > 0. Then there is a

8X e (0, s j with the following property. If P C Y is a closed subset homeomorphic to R r and

y0, z0 e P satisfy Z ( 0 ( j o z0), ax) < S l5 then there are pointsyx, zx e P so that

(42) d{z0, zx) = d{y9,yx) = d{y0, z0)

(44) ^mjTh), «i) < &i-

The proof requires:

Sublemma I.Z.I. — Suppose x,y, z e Y and Lx(y, z) = max(D(0(rji), Q(xz)) (cf. 3. \).Then x,y, z are the vertices of aflat (convex) triangle andyz e S v Y lies on the segment joining yxto a point v G 2^ Y, where 0(y) = Q(xz) and v and yx lie in a single chamber.

Proof of Sublemma 7.2.7. — Extend the geodesic segments ry, ~xz to geodesic rays x\x

and A;^2, £j edTi t sX. By hypothesis

Z.x(y, z) = max(D(6(^), Q(Tz))) = max(D(8(51), 0(^2))) = Z.rate(^, y .

So x\x £a determine a flat convex sector S. Note that yx a n d j | 2 lie in a single chamberof 1iv X since

Ly{x, Za)=n- Lv{lXi y = TC - max D(6(?1), 6(5,))

= minD(Ant(e(y), 6(5,)) = minD(6(jS:), 6(5,)).

Hence Axyz bounds a flat convex triangle T C S, and so yz lies on the geodesic segment

which has endpoints yx and y\%. •

Proq/" o/" Lemma 7.2.6. — Pick zx e P so that z0 zxC P, d(z0, zx) = d(y0, z0),

6(z0 %) = a, and z0 ^ e S ^ Y lies in a chamber antipodal to zoyo; similarly choose

yx e P so that j j ^ C P, </(jo,Ji) = <*(jo> *o)> 8(Jo^i) = Ant(a), and y~^x e SVo Y lies

in a chamber antipodal t o j 0 z0. Applying sublemma 7.2.7 we conclude that z0, y0, z1

are the vertices of a flat convex triangle, and y0 zx e SVo Y lies on the segment joining

y0 z0 to v eSy o Y where Q(v) = 6(z0 z-,) = a and v and y0 z0 lie in the same chamber.

In particulary0 zx a n d j o j x lie in antipodal chambers of SVo Y, so applying lemma 7.2.7

again, we find that Q(y1 zx) lies on the segment joining e(jy0 z-,) to 6(j1jyo) = a. jyx and zx

clearly satisfy the stated conditions since

and Ao(jo> «i) ^ /-^(Jo, «i) = TC — /L{%{y~T0), a) > TC - Sx > TT - sx. •

Corollary 7.2.8. — i«f a2 6 Amod ie a regular point, and let L, C, e2 > 0 be given. Thenthere are D2 > 0, S2 £ (0, s2) with the following property. If Q_ c X is an (L, C) r-quasiftat,andy0, z0 e Q, satisfy

(45) flT(j0, *„) > D,, Z(6( J o z0), a,) < S2

then there are points yx, zx e Q_ so that

(46) | d{z0, zj) - d{y0, z0) |, | d{yo,yx) - d{y0, z0) | < e2 d(y0, z0)

(47) 2,,(A» ^O)5 A0(j<» ZI) > ™ - e2

(48) ^ - ( e ( j ^ ) , a,) < 8,.

Proq/". — Let S2, X2 be the constants produced by the previous lemma with ax = a2,ex = e2. We claim that when y0, z o e Q and Z_(6(jy0 z0), a2) < S2 and d(y0, z0) issufficiently large, then there will exist points ylt zx satisfying (46), (47), (48). But thisfollows 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 yu zi e Q,

i ^ 1 such that the inequalities (45), (46), (47), (48) hold when we increment all the indices on

the fs and z's by i.

Lemma 7.2.10. — Fix |x > 0, and consider all configurations (y, z, F) where y, z e X ,/.(Q(yz), d Amod) > \x, and F C X is a maximal flat. Then there is a D3 such that the fractionof the segment yz lying outside the tubular neighbourhood NDa(F) tends to zero with

v (y, z, F) ^ mzx(d(y, F)jd{y, z), d(z, Y)\d{y, *)).

Proof. — Recall that the distance function d(F, .) is convex, so if the lemma werefalse there would be sequences yk, zk, wk e X, ¥kC X, with Z.(6(jyfc zk), d Amod) > |x,d(y, z) -> co, wk ey-Tk with d(wk,yk), d{wk, zk) > e d{yk, zk), vk{yk, zk> Fk) -> 0 butd(wk, Fk) -> oo. Let pk,qk,rkeFk be the points nearest yk, wk, zk respectively.By various triangle inequalities and property (39) from section 5.2 we have

^ ( A . A ) , Z«(r*, h) - 0 and Z(0(/^), 0(A1D), Z-(H%Tk), 8(J^))->0. There-fore if we set Rft = d(qk, wk) and take the ultralimit of ( R ^ - X , qk) we will get a confi-guration qa, waeX.a, an apartment FUCXM , and \x, %2 e dTlta XQ so that ?u is

the point in Fm nearest to wa, {qa lly qa) = co-lim(^T^, qk), {qa , qo) = t o - l i m ( ^ , g j ,ft). ( <o 52, ?w) = co-lim(a»,. *4, yfc). In particular, the rays

wk^1 and Wj. 2 fit together to form the geodesic oo-lim yk zk and Z.(0(^), 8 Amod)But this contradicts corollary 4.6.4. •

Corollary 7 . 2 . 1 1 . — HA; OC3 e Amod . TAm Mere are constants e4 , v4, D 4 such that if1. j j , ZjSX, i ^ 0 are sequences which satisfy (45), (46), (47), (48) (when subscripts are

incremented by i) with e2 < e4, </(jv0, z0) > D4 .2. 4 maximal flat F C X satisfies d(yk, F), d(zk, F) < v4 <i(jv&, zk) for some k.

Then d(yt, F), </(«,, F) < v4 rf(j>4) z,) /or a// 0 < i < ft.

Proo/". — If v4 is sufficiently small, then the trisection points y, 7 of any sufficientlylong segment^ C X with Z.(8(J5), a A ^ ) ^ fx, max(^(j, F)jd{y, z), d{z, F)ld(y, z)) < v4

will satisfy max{d(y,F)ld(y,7), d(7,F)ld(y)7)) < v4 by lemma 7.2.10. If we takee4 < v4 then Z.(0(j4 z4), S Amod) will be bounded away from zero and j ' 4 _ 1 , 2;i_1 willlie close to the trisection points of y% zi so corollary 7.2.11 follows by induction onk — i. •

Proof of Theorem 7 .2.1.

Step 1. — Fix a4 e Amod, and let e5, vB, D6 be the constants produced by corol-lary 7.2.11 with a3 = a4. Let D6, S6 be the constants given by corollary 7.2.8 witha2 = a4, s2 = s6. Finally, let D7 be the constant produced by corollary 7.2.5 witha = a4, s = min(§6, 1/2). Setting D8 = max(DB, D6, D7), for each y0 e Q, we may finda z o e Q w i t h D8<^(jy0 , ^0) < 2 D8 so that Z.(0(jyo, z0), a4) < 86 (by corollary 7.2.5).By corollary 7.2.9 we may extend the pair y0, z0 e Q to a pair of sequences yt, zt

satisfying (45)-(48) with a2 = a4, s2 = e6. Then any maximal flat F C X with

d(j>k> F)> d(zh, F ) < \ diyk> h) f o r s o m e ° < k < °° satisfies d(yt, F), rffo, F) < v6 d(yit zt)

for all 0 < t < & by corollary 7.2.11; in particular

(49)

We may assume in addition that s5 is small enough so that

(50) 2d(yi_1,zi_1)<d(yi,zt)<4d(yi_1,zt_1), and

(51) d(yiiyi_x), d(Zi, Zi_,) < 2 d{y,_x, Xi_x).

It follows that

(52) max(rfU,j,0), d(zt,y0)) < 2 d(yt, zt)

for all i.

Step 2. — Fix q e Q,and set v6 = vB/16. For each R pick a covering of Ba(R) n Q,by v6 R-balls { Bj,.(v8 R) } with minimal cardinality; the cardinality of this coveringcan be bounded by r and the quasiflat constants (L, G). For each pair pi, pf of centerspick a maximal flat containing them, and denote the resulting collection of maximalflats by .FB .

Claim. — Ify0 e Q , then d(y0, UFe5rB F) < 2v5 D8/or sufficiently large R.

Proof of claim. — We will use the sequences^, zi constructed in step 1 and esti-mate (49). Take the maximal i such that yi} zi eBa(R). Then

R - d{q,y0)

- d{q,y0)) by (52)

by (50).

Since SF-% contains a maximal flat F with

>1,F),<f(*1,F)<veR =

R

Therefore for sufficiently large R there is an F e J ^ and k such that

so d(y0, F) < 2v5 Dg as claimed. •

Proof of Theorem 7.2.1 concluded. — We may now take a convergent sub-sequence of the ^"E'S, and the limit collection J5" satisfies Q,CN2D8(Uj.e#-F) andcard(J?r) < lim sup card (^j) which is bounded by r and (L, C). D

Proof of Corollary 7.2.2. — By theorem 7.2.1 there is a finite collection !F ofmaximal flats so that Q_ lies in a finite tubular neighbourhood of UF e & F. The limitset of each F e & is its Tits boundary STlts F, which is an apartment of STits X. Theunion of these apartments gives us a finite subcomplex 3? C 8Tits X which is a unionof closed Weyl chambers.

Clearly IimSet(QJ s ^ ; we will show that if E, e IimSet(QJ then \ lies in aclosed Weyl chamber GC LimSet(QJ. We have qk e Q_ such that * ^ -* *5 in thepointed Hausdorff topology.

Consider Uyg^F . Any ultralimit w-lim(R^"1- (UF6i35- F), *) is canonicallyisometric to the Euclidean cone over 'S. The set w-lim(R^"1-Q^, •*) embeds in

1- (U F e # - F), •) as a bilipschitz copy of E r; by the discussion in section 6.2,^ Q , , *) is the cone over a collection of closed Weyl chambers in ^ . In par-

ticular co-lim *q~k = *a qa lies in a closed Weyl chamber contained in <o-lim(R^1.Q-, *) ,so the corresponding Weyl chamber of & is contained in LimSet(Q_), and it contains £. •

8. QUASI-ISOMETRIES OF SYMMETRIC SPACESAND EUCLIDEAN BUILDINGS

In this section our goal is to prove theorems 1.1.2 and 1.1.3 stated in theintroduction.

Let X, X', and <J> be as in theorem 1.1.2. By corollary 7.1.5, carries apartmentsclose 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 flatin X and s/(F) is the collection of apartments containing F, then n A 6 J ^ ( r ) N E (A) C ND ( E )(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 transvections along

geodesies in F preserve all the flats containing F. Hence, if there is a sequencexn e r iA e r f ( P ) NE(A) with d(xn,F) tending to infinity, then we may assume withoutloss of generality that the nearest point to xn on F is a given point p. The segments p~x~a

24

subconverge to a ray p\ which lies in n^e^F) NR(A) and is orthogonal to F. Sincefor each apartment A s j / ( F ) , we have p e A and the ray />£ remains in a boundedneighbourhood of A, it follows that/>£ C flxes?m F. Hence r i A e ^ ( F ) F contains a (k + 1)-flat, which is a contradiction.

Assume now that X is an irreducible thick Euclidean building with cocompactaffine Weyl group. Consider a point x e X \ F and let p e F be the nearest point in F.Then u : = px e Y,v X satisfies Z-P{u, Sp F) > 7t/2. We pick a chamber G in S p X con-taining u and choose a face a of G at maximum distance from u. Denote by v the vertexof G opposite to a. By our assumption, diam(Amod) < TT/2 and therefore v $ ~LV F. SinceF is a finite intersection of apartments, lemma 4.1.2 implies S p F = DA6J^ (F)S1,Aand there is an apartment A with F C A C X and v $ 1lp A. Sp A is then disjoint fromthe open star of v, and so d{u, Sj, A) ^ d(u, a) > a0 > 0 where a0 depends only on thegeometry of Amod. If x eNE(A) then angle comparison implies that d(x, F) ^ R/sin <x0

and our claim holds with D(R) = R/sin <x0. 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

at finite Hausdorff distance from <1>(A). There are constants D0(L, C, X, X') and D(L, C, X, X')so that if F = f l A - , F A C X is a singular flat, then

1. ( D ( F ) c n A D P N D o ( A ' ) ,2. The Hausdorff distance ds{$(F), f l A : ) F NDo(A')) < D,3. There is a singular flat F ' C D A : ) F N D o (A ' ) with da(<t>(F), F') < D.

In particular, two quasi-isometries <!>1, <J>2 : X ->• X ' inducing the same bijection on apartmentsinduce the same map of singular flats up to 2D-Hausdorff approximation.

Proof. — Let F and ^/(F) be as in the previous lemma. By corollary 7.1.5, forevery apartment A £ X, <p(A) is D0-HausdorfF close to an apartment in X' which wedenote by A'. Thus <p(F) C f lA e r f W ) NDo(A').

Sublemma 8 . 1 . 3 . — For each d^ Do there exists a constant Dx = D1(L, C, d) > 0with the property that r i A e ^ ( F ) Nd(A') lies within Hausdorff distance Tdxfrom <p(F).

Proof. — Pick a quasi-inverse cp"1 of 9. For each point y e riAej^ (F ) Nd(A') andeach A es/(F), <$~1y is uniformly close to qj"1 A'. But 9""1 A' is uniformly Hausdorffclose to cp~1 9A and therefore to A. Lemma 8.1.1 implies that Y has uniformly boundeddistance from F. •

Proof of Proposition 8.1.2 continued. — Fixing Ao e cj/(F), we conclude thatG : = (riAG^ (F ) N2Do(A')) n Ao is a convex Hausdorff approximation of <p(F).

Sublemma 8 . 1 . 4 . — Let C C E' be a convex subset which is quasi-isometric to E*. Then

C contains a k-dimensional affine subspace.

Proof. — Fix q e C and let C £ C be the convex cone consisting of all completerays starting in q and contained in G. For any sequence Xn -> 0 of scale factors, theultralimit co-lim(Xn-C, q) is isometric to C. Therefore G is homeomorphic to Efc andhence isometric to E*. D

Proof of Proposition 8.1.2 continued. — It follows that cp(F) is uniformly close toa flat F in X'. Since cpw carries singular flats to singular flats, dTita F is a singular spherein 0Tits X'. X' has cocompact affine Weyl group, so F lies within uniform HausdorfFdistance from a singular flat F'. D

8.2. Rigidity of product decompositionand 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) x a = E " x n x i Q , x ; = E-' x n x;B

where the X j u , X^M are irreducible thick Euclidean buildings. They have the propertythat every point is a vertex and their affine Weyl group contains the full translationsubgroup, in particular the translation subgroup is transitive. We are in a position toapply theorems 6.4.2 and 6.4.3: The Euclidean de Rham factors of X and X' haveequal dimension, n = n', and X, X' have the same number of irreducible factors. Afterremumbering the factors if necessary, there are homeomorphisms (<pu)4: Xiw -> X,'usuch that

where pt: X -> X4 and p[ : X' -> Xj' are the projections onto factors. Now let F bea singular flat which is contained in a fiber of/^. By proposition 8.1.2, cp(F) is uniformlyHausdorfF close to a flat F' C X'. Since F^ C X^ is contained in a fiber of p'ia, F' mustbe contained in a fiber of/>,'. Any two points in a fiber p^1(xi), xt eX4 , are containedin some singular flat FC^,"1^,) and consequently <p carries fibers of p{ into uniformneighbourhoods of fibers of p[. Since an analogous statement holds for a quasi-inverseof 9, we conclude that X,' so that

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 thatX' is also irreducible, with rank(X) = rank(X').

8.3.1. Quasi-isometries are approximate homotheties

We recall from proposition 7.1.5 that <& carries each apartment A in X uniformlyclose to a unique apartment in X' which we denote by A'. We prove next that in ourirreducible higher-rank situation the restriction of to A can be approximated by ahomothety. As a consequence, the quasi-isometry <J> is an almost homothety. Thisparallels the topological result in section 6.4.4.

Proposition 8 . 3 . 1 . — There are positive constants a = a(O) and b = #(L, G, X, X')such that for every apartment A C X exists a homothety WA : A -> A' with scale factor a which

approximates O|A up to pointwise error b.

Proof. — If we compose O|A with the projection X' -»• A', we get a map YA : A -> A'which, according to proposition 8.1.2, carries walls to within bounded distance ofwalls. Parallel walls in A are carried to Hausdorff approximations of parallel wallsin 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 3.3.2exist f + 1 singular half-spaces in A which intersect in a bounded affine r-simplex withnon-empty interior. Consider the collection ^ of hyperplanes in A which are parallelto the boundary wall of one of these half-spaces. Any r pairwise non-parallel hyperplanesin # lie in general position, i.e. intersect in one point. Hence we may apply lemma 8.3.3below to the collection <jf and conclude that WA is within uniform finite distance ofan affine transformation *FA : A -> A'. Since <J>M is a homothety on asymptotic conesby the discussion in section 6.4.4, it follows that *FA is a homothety: For suitable positiveconstants aA and b we therefore have

I <*(^A(*I), ^ A ( * 2 ) ) - «A d(xi> *i) I < * V *1} x2 e A

and b depends on L, C, X, X' but not on the apartment A. To see that the constant aA

is independent of the apartment A note that for any other apartment A t C X thereis a geodesic asymptotic to both A and A1. It follows that aA = aA. •

Corollary 8 . 3 . 2 . — There are positive constants a = a(O) and b = b(L, G, X, X')such that the quasi-isometry <& : X -> X ' satisfies

) , © ( * , ) ) - a.d{Xl, x 2 ) \ ^ b V Xl, x2 e X .

Here L" 1 < a < L.

Proof. — This follows from the previous proposition, because any two points in Xlie in a common apartment. D

Lemma 8 . 3 . 3 . — For n > 2, let <x0, . . . , aB e (R")* be a collection of linear functional

any n of which are linearly independent, and let be the collection of affine hyperplanes { a,""1 (c) }c e „ .There is a function D(C) with limc_,.0D(C) = 0 satisfying the following: If 9 : R* - > R"

is a locally bounded map such that for all H eJf^., <p(H) C NC(H') for some H ' eJf , , then

there is a an affine transformation ip0 with scalar linear part which preserves the hyperplane families J^.such that d(<p, <p0) < D(G).

Proof. — After applying an affine transformation if necessary we may assumethat oc0 = S"= 1 xt, a.j = x} for 1 < j< n, and cp(O) = 0. There is a C 2 e R such thatthe image of each &-fold intersection of hyperplanes from U, J^t lies within the C2 neigh-bourhood an intersection of the same type. In particular, for each 1 j < n, 9 inducesa (C3, s3) quasi-isometry <p;. of the j-th. coordinate axis, with 9^(0) = 0. It suffices toverify that each <p} lies at uniform distance from a linear map since 9 lies at uniformdistance from n " = 1 <p}. Also, it suffices to treat the case n = 2 since for each 1 j < nwe may consider the (C4, s4) -quasi-isometry that 9 induces on the xi xi coordinateplane, and this satisfies the hypotheses of the lemma (with somewhat different constants).

We claim there is a C5 such that for y, z in the first coordinate axis, we haveI 91(j> + z) — (9i(j) + 91(2)) I < G5. To see this first note that when C equals zerothe additivity can be deduced from a geometric construction involving 6 lines and 6of their intersection points. When C > 0, the same construction can be performed withuniformly bounded error at each step.

By lemma 8.3.4 below, ^1 and analogously 9,- lies at uniform distance from alinear map. •

Lemma 8 . 3 . 4 . — Suppose ty : R -> R is a locally bounded function satisfying

+ z) — ^{y) ~ i>iz) I < D for ally, z e R . Then | ty(x) — ax \ < D for some a e R.

Proof — Since | ^(2") — 2^(2"-1) | < D, the sequence (4»(2")/2") is Gauchy andconverges to a real number a. Let x> 0 and choose numbers } B e N and rn e R withI rn I < x such that 2" = qn x + rn. Then

and hence, using that ^ is locally bounded,

<K2")o» - -hr <K*) -

YVn) X

2"- + 0

- -*

2"D.

When n tends to infinity, we obtain in the limit

I ax — <K*) I < D .

Similarly, there is a real number a_ such that for all x < 0 we have | a_ x — <\i(x) \ < D.Since | <\>(x) + ty(— *) | < D + | 4*(0) |, it follows that a = a_. n

Proof of Theorem 1.1.3 concluded. — By corollary 8.3.2 we may scale the metricon X' by the factor I/a so that $ becomes a (1, A/a) quasi-isometry. Applying propo-

sition 2.3.9 we conclude that induces a map 8X O : 8X X -> 8m X' which is a homeo-morphism of geometric boundaries preserving the Tits metric. By the main result of 3.7,8X O gives an isomorphism of spherical buildings 8X O : (8Tlta X, Amod) -> (dTit8 X', A ^ ) ,after possibly changing to an equivalent spherical building structure on dTits X'.Consequently, for every 8 e Amod, 8X <J> maps the set S"1^) C 9Tit8X to the corres-ponding set 6'~1(8) C dTlts X', and O 6_i(8) is a cone topology homeomorphism. When8 is a regular point, the subsets 6 - 1 (S)caT l t eX and O'-^S) C 8TltaX' are eithermanifolds 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. Thereforeeither X and X' are both symmetric spaces of noncompact type, or they are bothirreducible Euclidean buildings with Moufang boundary. In the latter case we aredone 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 8m O0 = 8m <X>. By proposition 8.1.2,rf(0(o), ®0(v)) < D for every vertex v e X, and since the affine Weyl group of X iscocompact the vertices are uniform in X, and so we have d(Q>, O0) < D'. Hence <E>0 isan isometry. •

8.3.2. Inducing isometries of ideal boundaries of symmetric spaces

We consider a symmetric space X of non-compact type and denote by G theidentity component of its isometry group.

Sublemma 8 .3 .5 . — Let F C X be a maximal flat and let 7tF: X -> F be the nearestpoint retraction. Given a compact set K C Int(Amod) and e > 0, there is a 8 > 0 such that ifp G X, xeF, Q(px) e K, and /.„(*, TCp(p)) > TC/2 — 8, then d{p, F) < s.

Proof. — Note that as q moves from/> to %(/>) along the segment 7Cj,(/>) p, /-v{x,increases monotonically. If the sublemma were false, we could find a sequence pk e X,* f ceF so that ZP4(*fc, Mpk)) -> TC/2 and d(ple,F)>z. Since Z.^H){xk, pk) = rc/2,triangle comparison implies that | pk TzF(ph) |/| pk xk | -> 0. Hence by taking qk epk Kv(pk)with d(qk,F) = s we have LXk{pk,qk) ->0, so dAmoJQ(qk xk), K) ->0. Modulo thegroup G, we may extract a convergent subsequence of the configurations (F, qk xk)getting a maximal flat F, a point q^ with d(qm, F) = s, and xm e 8m F such that-4oo(*oo> ip(?«,)) = w/2» and 6 ^ ) e K. This is absurd. D

Sublemma 8 . 3 .6 . — Let F4 be a sequence of maximal flats in X so that 8X F4 ->• 8X FMJACT? F M fl maximal flat, i.e. for each open neighborhood U of 8^Y in 8X X with respect to the

cone topology, 8m F{ is contained in Vfor sufficiently large i. Then F4 -> F in the pointed Hausdorff

topology.

Proof. — Let \, t\ e 8X F be antipodal regular points and choose points ^ , i]j 6 SB Fjso that £j -> \ and t^ ->• i\. Then for x e F we have /.„(£,, vjj -»• 7t and consequently

/•„(%,. x, !-,) -> n/2, A,(%,. #, >)i) -*• TI/2- Applying sublemma 8.3.5, we conclude thatd(x, Ft) -> 0. The claim follows since this holds for all x e F. •

Lemma 8 . 3 .7 . — Let d^ : G ->Homeo(S00 X) be the homomorphism which takes each

isometry to its induced boundary homeomorphism. Then dx is a topological embedding when

Homeo(^00 X) is given the compact-open topology.

Proof. — The homomorphism dm is continuous, because the natural action of Gon dm X is continuous. To see that dm is a topological embedding, it suffices to showthat if ^ e G is a sequence with #„(&) -+ e e Homeo(^00 X), then gi -*• e e G. Let xbe a point in X and choose finitely many (e.g. two) maximal flats F1, ..., Fk withF1n ... n Fj. = { x }. Since 0J&) -> e e Homeo(aoo X), «»«, & F, converges to dn F,in the sense that for each open neighborhood U3- of dm F3- in 8X X with respect to thecone topology, dK gi F3- is contained in U^ for sufficiently large i. By the previoussublemma we know that & F}- ->• F3- in the pointed Hausdorff topology, n

Proposition 8 . 3 .8 . — Let X and X ' be irreducible symmetric spaces of rank at least 2.Then any cone topology continuous Tits isometry

| : aTits X -> 8TltB X '

w induced by a unique homothety *F: X -> X ' .

Proof. — We denote by G (resp. G') the identity component of the isometry groupof X (resp. X'). By lemma 8.3.7 the homomorphisms 8X : G -> Homeo(900 X) andd'n : G' -> Homeo(S0O X') are topological embeddings, where Homeo(500 X) andHomeo(500 X') are given the compact-open topology. According to [Mos, p. 123,cor. 16.2], conjugation by ty carries 3fflG to d'^G'. Hence <\> induces a continuousisomorphism G ->• G'. Such an isomorphism carries (maximal) compact subgroupsto (maximal) compact subgroups and it is a classical fact that the induced map T : X -> X'of the symmetric spaces is a homothety. The isometry ty and the induced isometrydTlte T at infinity are G-equivariant with respect to the actions of G on dTitg X andaTIts X' and we conclude that 8Tite W = <J>. •

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 G so that for every (1, A) quasi-isometry O : X -*• X ' there is an

isometry O0 : X -* X ' with d($>, <3>0) < G.

The statement is false for (1, A) quasi-isometries between trees.

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, Amod) be a spherical building. Attached to each root (i.e. half-apartment)in B is a root group Ua £ Aut(B, Amod) (see 3.12). Remarkably, when B is irreducibleand has rank at least 2, the Uo's—and consequently the group G c Aut(B, Amod)generated by them—act canonically and isometrically on any Euclidean buildingwith Tits boundary B (see 4.10). Now let (B, Amod) = (eTit3 X, Amod). If<& : X -*• X' is an (L, A) quasi-isometry, then by 2.3.9 we get an induced isometryaTltB 0> : aTits X -* aTit8 X', so the group G s Aut(B, A m J acts on STlt8 X, 8TitB X', andhence on X and X'. By comparing images of apartments (and using the quasi-isometry <&), one sees that a subgroup K c G has bounded orbits in X if and only ifit has bounded orbits in X'. Because B is Moufang (3.12) the maximal bounded sub-groups MC G pick out "spo ts" J M e X and B H E X ' (proposition 4.10.6), and theresulting 1-1 correspondence between the spots of X and the spots of X' determinesa homothety O0 : X -> X' with STits $ 0 = 8Tlts ®.

Proof of Theorem 8 .3 .9 . Step 1. — Reduction to the irreducible case.

Lemma 8 .3 .10 . — Every (1, A) quasi-isometry cp : E r - > E r lies within uniform distance

of a homothety.

For every distance function d: E r -> E ' the function d o 9 lies within uniformdistance of a distance function. By taking limits we see that for every Busemann functionb : E r -> Er, b o cp is uniformly close to a Busemann function. But the Busemann functionsare affine functions, so <p is uniformly close to an affine map cp0. Obviously cp0 is anisometry. •

By corollary 7.1.5, there is a constant D(A, X, X') such that the image of everyapartment A C X is D Hausdorff close to an apartment A' C X'. Composing O A withthe projection onto A' we get a map which is uniformly close to an isometry *FA : A -> A'.Hence if F C A is a flat, then O(F) C X' is uniformly Hausdorff close to the flat TA(F) C A'.Therefore we may repeat the reasoning of 8.2 to see that if X = IIXj, X' = IIX,-are the decompositions of X and X' into thick irreducible factors, then after reindexingthe factors XJ there are (1,A) quasi-isometries O i : X{ -> X,' so that <J> is uniformlyclose to n $ 4 (A depends only on the quasi-isometry constant of O and X, X'). Hencewe are reduced to the irreducible case.

Step 2. — The buildings X and X ' are irreducible. The affine Weyl groups W ^ , W^.fof X, X' are either finite or cocompact, since their Tits boundaries are irreducible.If Wgjj is finite then it has a fixed point, so all apartments intersect in a point p e Xand X is a metric cone over 9Tlt8 X. If a e Amod is a regular point, then 6-1(a) C STit8 Xis clearly discrete in the cone topology. On the other hand, if W ^ is cocompact then

8-1(a) C dTiU X is nondiscrete since any regular geodesic ray p\ C A can branch offat many singular walls. Since O induces a homeomorphism of geometric boundaries8X 0 : dm X -> dx X' by 2.3.9, and this induces an isomorphism of spherical buildingsdTitg : dTita X -> dTita X', either X and X' are both metric cones, or they both havecocompact affine Weyl groups. If they are both cones, we may produce an isometryO0 : X -> X' by taking the cone over dTita O : dTita X -> dTit8 X'. This induces the samebijection of apartments as <&, and lies at uniform distance from <f> 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(5Tite X) ~ Aut(3Tita X')

be the group generated by the root groups of dTitB X, we get actions of G on dTjtg X,8TltB 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) = { ^ | ^ e K } C X thenK has a bounded orbit in X'.

Let p e X be a vertex, let S^v be the collection of apartments passing through p,and let &K{p) = UgeK&r

tp. &•&») is a K-invariant collection of apartments in X,and when R > Diam(K(/>)) we have p e t\Ke!rHp) NB(A). Let <J>(J*;) and ®(&K(t))denote the corresponding collections of apartments in X'. Then ^{^K. .) is K-invariant,and $ ( / ) s HA, 6q>(^K , NE + 0 (A'), where Gx is a constant such that for every apart-ment A C X , the Hausdorff distance <fH(<D(A), A') < Cx. By proposition 8.1.2,fW(JvNE + Ci(A') is bounded. Thus n A , e W l ( t ) ) NE + Ci(A') is a nonemptyK-invariant bounded set. •

Proof of Theorem 8.3.9 continued. — By proposition 4.10.6 we now have a bijection

Spot(0>) : Spot(X) -> Spot(X')

between spots in X and X' via their correspondence with maximal bounded subgroupsin G. Moreover by item 2 of proposition 4.10.6 for every apartment AC X, wehave Spot($) (Spot(A)) = Spot(A') where A' C X' is the unique apartment with

A ) - Since by item 3 of proposition 4.10.6

|8pot(A): Spot(A) - • Spot(A')

is a homeomorphism with respect to half-apartment topologies we see that X is discreteif and only if X' is discrete.

Case 1: Both X and X ' are non-discrete, i.e. their affine Weyl groups have a dense orbit.

In this case Spot(A) = A, Spot(A') = A', and Spot($) |A : A -> A' is a homeomorphism25

since the half-apartment topology is the metric topology. By item 3 of proposition 4.10.6Spot(3>) |A maps singular half-apartments H C A with dTlte H = a to singular half-apartments Spot(O) (H) C A' with 3Tlte(Spot(O) (H)) = a,,lte(a). By consideringinfinite intersections of singular half-apartments with Tits boundary a C 3TItg A, itfollows that Spot(<J>) carries all half-spaces H C A with dTits H = a to half-spacesSpot(<D) (H) with aTita(Spot($) (H)) = dTltB<D(a). By considering intersections ofhalf-spaces H ± with opposite Tits boundaries, we see that Spot(<&) carries hyperplaneswhose boundary is a wall m C dTito A to hyperplanes in A' with boundarySTjtlO(ff!)Cailt!A'. By section 6.4.4 it follows that % = Spot(O) : X -* X' is ahomothety and 8Tlta O0 = 9Tita <1>.

Case 2: X and X' are both discrete. In this case A and A' are crystallographic EuclideanCoxeter 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 singularhalf-apartment, then there is a unique singular subspace or singular half-apartmentS' C A' so that Spot(<5) (S n Spot(A)) = S' n Spot(A'). k + 1 spots s0, ...,ske Spot(A)are the vertices of a ^-simplex in the simplicial complex if and only if they do not liein a singular subspace of dimension < k and the intersection of all singular half-apartmentscontaining { s0, ..., sk } contains the k + 1 spots st. Hence

Spot(O)|8potA : Spot(A) -* Spot(A')

is a simplicial isomorphism and hence is induced by a unique homothety A -> A'. Itfollows that Spot(O) : Spot(X) -> Spot(X') is the restriction of a unique homothety<D0 : X -* X' with aTita % = aTita 0).

Since vertices are uniform in X, we may apply proposition 8.1.2 to concludethat in both cases <f(O0, <&) < D'(L, C, X, X'), forcing <D0 to be an isometry. Q

9. AN ABRIDGED VERSION OF THE ARGUMENT

In this appendix we offer an introduction to the proof of Theorem 1.1.2 via thespecial case when X = X' = H2 X H2.

Step 1. — The structure of asymptotic cones co-lim(Xj(H2 x H2) , zj. Readers unfamiliarwith asymptotic cones should read section 2.4. By 2.4.4, any asymptotic coneto-lim(Xj H2, xt) is a GAT(K) space for every K, SO it is a metric tree; since there arelarge equilateral triangles centered at any point in H2, the metric tree branches every-where. The ultralimit operation commutes with taking products, so one concludes thatw-lim(Xi(H2 X H2), z() ~ w-lim(Xi H

2, xt) X w-lim(Xi H2,jj) where zi = xi X y{ and

X denotes the Euclidean product of metric spaces. So any asymptotic cone of H2 x H2

is a product of metric trees which branch everywhere.

Step 2 . —• Planes in a product of metric trees are " locally finite ". F o r i = 1 ,2 le t T^be a metric tree. For simplicity we assume that geodesic segments and rays are extendible

to complete geodesies. Since the convex hull of two geodesies in a metric tree is containedin the union of at most 3 geodesies, the convex hull of two 2-flats y< X 84 C Tx X T2

is contained in at most nine 2-flats. Section 6 may now be read up to the paragraphafter lemma 6.2.1, replacing the word " apartment " with " 2-flat ", and corollary 4.6.8with the observation above. Hence every topologically embedded plane in Tx X T2

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 Ti branch everywhere:for every * e T j , T4\x has at least 3 components. Let P C Tx X T2 be a topologicallyembedded plane, and let z — x X y e P. We know that there are finite trees T{ C Tt

with z e T j X T j C T j X T , so that Bz(r) n PC Bs(r) n (Tx X T2). Shrinking r ifnecessary, we may assume that Tx and T2 are cones (x e Tx and y e T2 are the onlyvertices). Elementary topological arguments using local homology groups show thatBz(r) n P coincides with B8(r) n (UQ,^, where each Qi C 1 \ X T2 is a quarter planewith vertex at z, i.e. a set of the form y x S C Tx x T2 where y C Tx (resp. 8 C T2) isa geodesic leaving x (resp. y).

Say that two sets Sx, S2 C Tx X T2 have the same germ at z if Sx n U = S2 n Ufor some neighborhood U of z. We see from the above that for every z e P, P has thesame germ at z as a finite union of quarter planes. Moreover, since the intersectionof two quarter planes Q1} Q_2 with vertex at z either has the same germ as Q,4, the samegerm as a horizontal or vertical segment, or the same germ as { z }, it follows that a setS C Tx x T2 has the germ of a quarter plane with vertex at z if and only if it has thesame germ as a two-dimensional intersection of topologically embedded planes, andis minimal among such. Hence we have a topological characterization of 2-flats andvertical/horizontal geodesies: a closed, topologically embedded plane P C T1 X T2 isa 2-flat if for every z e P, P has the same germ at z a the union of four quarter planeswith vertex at z; a closed connected subset S C T1 X T2 is a vertical or horizontalgeodesic if for every z e S, S has the same germ at z as the boundary of two adjacentquarter planes with vertex at z. From this one may easily recover the product structureon T1 X T2 using only the topology of Tx x T2 . Hence a homeomorphism Tx x T2 preserves the product structure (although it may swap thefactors, of course).

Step 4. — Quasi-isometries of H 2 x H 2 preserve the product structure. Let

0): H2 x H2 -* W x H2

be a quasi-isometry. If z, z ' e f f x H2, let 6(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 limr^xf{r) = 0 sothat if z, z' are horizontal, then \ 8(0>(z), >(z')) — W4 I > W4 — / W -

1 % BRUCE KLEINER AND BERNHARD LEEB

Proof. — If not, we could find a sequence zf, z[ e f f x f f of horizontal pairsso that Xr1 = d(z,, z[) = oo and lim sup,^,, | 6(0)(zj), $ « ) ) — TC/4 | < TC/4. Thenzus z'a e <a-lim(\(H* X H2), z j is a horizontal pair with 8(<DU(*J» * „ « , ) ) * 0, TC/2.This contradicts step 3. D

Since any two horizontal pairs zx, z[ and z2, z'2 may be joined with a continuousfamily zt, z\ of horizontal pairs with min d(zt, z't) > rain.(d(z1} z[), d(zz, z'2)), we seethat for horizontal pairs z, z', the limit limdu „.)_*„, 6(<l)(z), 5>(z')) exists and is either 0or TC/2. We assume without losing generality that the former holds.

Hence as y e H2 varies, the compositions H2 -> H2 X {y } -5- H2 X H2 5- H2 arequasi-isometries with quasi-isometry constant independent of y, and they lie at finitedistance from one another. It follows that they lie at uniform distance from one another,and so O preserves the fibers of p1 up to bounded Hausdorff error. Repeating this argu-ment for p2 we see that <1> is within uniform distance of a product 3>x X <1>2 of quasi-isometries.

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B. K.,Department of Mathematics,University of Utah,Salt Lake City, UT 84112-0090.email: [email protected].

B. L.,Fachbereich 17: Mathematik,Universitat Mainz,Staudingerwez 9,D-55128 Mainz,email: [email protected].

Manuscrit regu le 8 avril 1996.

publications mathematiques de l'i.h.e - ….pdfin contrast to this, homeo-morphisms of rank 1...

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