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Some remarks on proper actions, proper

metric spaces, and buildings

Linus Kramer ∗

December 2, 2021

Abstract

We discuss various aspects of isometric group actions on proper metric spaces.As one application, we show that a proper and Weyl transitive action on a euclideanbuilding is strongly transitive on the maximal atlas (the complete apartment system)of the building.

The purpose of this article is to supply references and complete proofs for some usefulfacts about proper group actions on proper metric spaces. These results are then appliedto group actions on metric simplicial complexes, and in particular to group actions onbuildings. Some of these results are known as folklore, though it is in some cases not soeasy to find references or reliable proofs.

The first section contains a discussion of proper maps and proper actions of topologicalgroups in general. The second section discusses isometry groups of metric spaces, andin particular isometry groups of proper metric spaces. The third section applies theseresults to Mκ-simplicial complexes, which have become a central tool in geometric grouptheory. The fourth section introduces buildings (in their various manifestations). One(reassuring) result for buildings is that all reasonable topologies on their automorphismgroups coincide. The fifth an final section contains result about various types of transitiveactions on buildings. For the case of proper actions, these transitivity conditions are shownto be essentially equivalent.

All topological spaces and groups are assumed to be Hausdorff spaces.

∗Funded by the Deutsche Forschungsgemeinschaft through a Polish-German Beethoven grantKR1668/11, and under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Munster:Dynamics-Geometry-Structure.

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1 Proper maps and proper actions

A continuous map f : X −→ Y is called proper 1 if it is closed and if preimages of pointsare compact [6, I.§10.2]. Then the preimage of every compact subset is compact. Propermaps are also called perfect maps [15, §3.7]. If Y is a k-space (a subset B ⊆ Y is closedif and only B ∩K is compact for every compact subset K ⊆ Y ), then a continuous mapf : X −→ Y is proper if and only if the preimage of every compact subset B ⊆ Y iscompact [15, 3.7.18]. We note that every first countable space, every locally compactspace and every CW complex is a k-space.

We collect some facts about these maps which can be found in Bourbaki [6] andEngelking [15].

Facts 1.1. If Y is locally compact and if f : X −→ Y is proper, then X is also locallycompact [15, 3.7.24]. If f : X −→ Y is a proper map, if A ⊆ X is a closed subset, ifB ⊆ Y is any subset, and if f(A) ⊆ B, then the restriction-corestriction f |BA : A −→ B isalso proper. Similarly, the restriction-corestriction f |Bf−1(B) : f

−1(B) −→ B is proper [15,

3.7.6]. The cartesian product of proper maps is again a proper map [15, 3.7.9].Suppose that

X Y

Z

f

hg

is a commutative diagram of continuous maps. If f and g are proper, then h is alsoproper. If h is proper, then both f and the restriction g|f(X) : f(X) −→ Z are proper [15,3.7.3,3.7.5].

The following observation turns out to be useful.

Lemma 1.2. Let f : X × Y −→ Z be a continuous map. If Z × Y is a k-space, then thefollowing are equivalent.(i) The map F : X × Y −→ Z × Y that maps (x, y) to (f(x, y), y) is proper.(ii) For all compact subsets C × B ⊆ Z × Y , the set

AB,C = {a ∈ X | there is b ∈ B such that f(a, b) ∈ C}

is compact.

In general, (i) implies (ii).

1This is Bourbaki’s notion of properness and the reader should be warned that other, different notionsof proper maps appear in the literature.

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Proof. Suppose that (i) holds. Then the set

K = {(x, y) ∈ X × Y | f(x, y) ∈ C × B and y ∈ B} = {(a, b) ∈ X ×B | f(a, b) ∈ C}

is compact, and so is its projection prX(K) = AB,C .Suppose that (ii) holds. Let L ⊆ Z × Y be compact. We have to show that the

preimage of L in X × Y is compact. Put B = prY (L) and C = prZ(L). Then C × B iscompact and contains L. The preimage of C × B in X × Y is closed and contained incompact set AB,C × B. Therefore the preimage of L is also compact.

If in addition Z = Y holds in Lemma 1.2, then it suffices that AB,B is compact for allcompact B ⊆ Y , because B × C ⊆ (B ∪ C)× (B ∪ C).

Proposition 1.3. Suppose that F : X × Y −→ Z × Y is a proper map of the formF (x, y) = (f(x, y), y). Then the following hold.(a) For each y ∈ Y , the set X(y) = f(X×{y}) ⊆ Z is closed and the map X −→ X(y)

that maps x to f(x, y) is proper and in particular a quotient map.(b) For all (y, z) ∈ Y × Z, the set Ay,z = {x ∈ X | f(x, y) = z} is compact.(c) Suppose that Y = Z, that the map f is open and that y ∼ f(x, y) is an equivalence

relation on Y . Then the quotient map q : Y −→ Y/∼ is open and Y/∼ is aHausdorff space.

Proof. The restriction-corestriction X×{y} −→ X(y)×{y} is proper by 1.1 In particular,it is a closed surjective map and (a) holds. For (b) we just note that Ay,z = prX(F

−1(z, y))is compact. For (c), let U ⊆ Y be open. Then q−1(q(U)) = f(X × U) is open, hence qis open. But then q × q is also open, and hence is a quotient map. The preimage of thediagonal D in Y/∼×Y/∼ is E = F (X×Y ) ⊆ Y ×Y , which is closed. Hence D is closed,and therefore Y/∼ is Hausdorff.

Definition 1.4. Suppose that G is a topological group and that G × X −→ X is acontinuous action. The action is called proper 2 if the map

G×X −→ X ×X, (g, x) 7−→ (gx, x)

is proper.

Proposition 1.5. Let G×X −→ X be a continuous action of a topological group.(a) If X is first countable (eg. metrizable) or locally compact, then the action is proper if

and only if for all compact subsets B,C ⊆ X the set GB,C = {g ∈ G | g(B)∩C 6= ∅}is compact.

2Again, the reader should be warned that other notions of proper actions appear in the literature, seee.g. [5, 18].

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(b) If the action is proper, then for every x ∈ X the G-orbit G(x) ⊆ X is closed, thestabilizer Gx is compact, and the natural map G/Gx −→ G(x) is a homeomorphism.The orbit space G\X is a Hausdorff space and the quotient map q : X −→ G\X isopen.

Proof. Claim (a) follows from Lemma 1.2 and Claim (b) is a consequence of Proposi-tion 1.3. See also [6, III.§4.2] for a direct proof.

Lemma 1.6. Let G × X −→ X and K × Y −→ Y be continous actions of topologicalgroups, let α : G −→ K be a morphism of topological groups, and let f : X −→ Y be anequivariant and proper map, so that the diagram

G×X X

K × Y Y.

α×f f

commutes. If the action of G is proper, then α is a proper map. In particular, α is closedand has a compact kernel.

Conversely, if α is closed with compact kernel, and if the K-action is proper, then theG-action is also proper.

Proof. We consider the commutative diagram

G×X X ×X

K × Y Y × Y.

α×f f×f

Assume that the G-action is proper. By 1.1, the map f × f is proper and therefore α× fis also proper. For x ∈ X , the restriction-corestriction G × {x} −→ K × {f(x)} is alsoproper. Hence α is closed, with compact kernel.

If α is closed and with compact kernel, then α is proper and therefore α × f is alsoproper. Then G × X −→ X × X is proper by 1.1, provided that K × Y −→ Y × Y isproper.

Corollary 1.7. Suppose that K × Y −→ Y is a proper action. If G ⊆ K is a closedsubgroup and if X ⊆ Y is a closed G-invariant subset, then the restricted action G×X −→X is also proper.

The following basic example shows that a continuous action with trivial stabilizersneed not be proper.

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Example 1.8. Let S = {u ∈ C | |u| = 1} and let a be an irrational real number. Then R

acts freely and continuously on S × S via (t, u, v) 7−→ (exp(√−1t)u, exp(

√−1at)v), and

the same holds for the discrete group Z ⊆ R. The orbits of these two actions are denseand therefore these actions are not proper.

A topological group acting properly on a locally compact space X is necessarily locallycompact by 1.1. Also, every compact (eg. finite) transformation group acts properly.However, proper actions are not necessarily related to local compactness.

Example 1.9. Let G be a metrizable topological group. Then the left regular action of Gon itself is proper. Indeed, if A,B ⊆ G are compact, then GA,B = {ba−1 | a ∈ A, b ∈ B}is compact.

We call a group action G×X −→ X cocompact if there is a compact set A ⊆ X thatmeets every G-orbit. The left regular action in Example 1.9 is cocompact and proper.

2 Isometry groups of proper metric spaces

Let (X, d) be a metric space. The isometric embeddings X −→ X form a monoid, whichwe denote by IE(X). The group of invertible elements in this monoid is the isometrygroup Iso(X) ⊆ IE(X). We note that every g ∈ IE(X) is a proper map,

The following is well-known [6, X.§3.5]. For the sake of completeness, we include theproof. For a subset Y ⊆ X and ε > 0 we put

Bε(Y ) = {x ∈ X | d(x, y) < ε for some y ∈ Y }.

Lemma 2.1. Let X be a metric space. Then the compact-open topology and the topologyof pointwise convergence coincide on IE(X). With respect to this topology, IE(X) is atopological monoid, the action IE(X)×X −→ X is continuous and Iso(X) is a topologicalgroup.

Proof. For A,U ⊆ X we put 〈A;U〉 = {g ∈ IE(X) | g(A) ⊆ U}. Then the sets 〈F ;U〉,where F is finite and U is open form a subbasis for the topology of pointwise convergence.The sets 〈K;U〉, where K is compact and U is open form a subbasis for the compact-open topology. Since every finite set is compact, this shows that the topology of pointwiseconvergence is coarser than the compact-open topology.

Let W ⊆ IE(X) be an open set in the compact-open topology, with g ∈ W . Thenthere are compact sets K1, . . . , Kn ⊆ X and open sets U1, . . . , Un ⊆ X , with

g ∈ 〈K1;U1〉 ∩ · · · ∩ 〈Kn;Un〉 ⊆ W.

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Since the Ki are compact, there exists ε > 0 such that g(Bε(Ki)) ⊆ Ui holds for i =1, . . . , n. Moreover, there are finite subsets Fi ⊆ Ki such thatKi ⊆ Bε/3(Fi). If h ∈ IE(X)with d(gp, hp) < ε/3 for all p ∈ F1 ∪ · · · ∪ Fn = F , then d(gq, hq) < ε holds for allq ∈ K1 ∪ · · · ∪Kn and thus h(Ki) ⊆ Ui holds for i = 1, . . . , n, that is,

g ∈⋂

p∈F

〈{p};Bε/3(f(p))〉 ⊆ 〈K1;U1〉 ∩ · · · ∩ 〈Kn;Un〉 ⊆W.

This shows that the compact-open topology is coarser than the topology of pointwiseconvergence. Hence both topologies agree. For the remainder of the proof, we work withthe topology of pointwise convergence.

Let p ∈ X and g, h, g′, h′ ∈ IE(X). If d(gp, g′p) < ε/2 and if d(hgp, h′gp) < ε/2, then

d(hgp, h′g′p) ≤ d(hgp, h′gp) + d(h′gp, h′g′p) < ε.

Therefore the composition IE(X)× IE(X) −→ IE(X) is continuous.Let (g, p) ∈ IE(X)×X . If h ∈ IE(X) and q ∈ X , with d(p, q) < ε/2 and d(gp, hp) <

ε/2, then d(gp, hq)) < ε, hence the joint evaluation map IE(X)×X −→ X is continuous.Suppose that g ∈ Iso(X) and that p ∈ X . Put q = g−1(p). If h ∈ Iso(X) with

d(hq, gq) = d(hq, p) < ε, then d(h−1p, g−1p) = d(p, hq) < ε. We have shown that h−1 ∈〈{p};Bε(g

−1p)〉, provided that h ∈ Iso(X) ∩ 〈{q};Bε(gq)〉. Hence inversion is continuousin Iso(X).

From now on we endow the monoid IE(X) with the topology of pointwise convergence.We recall that a metric space (X, d) is called proper if the Heine–Borel Theorem holds inX : a subset is compact if and only if it is bounded and closed. Proper metric spaces arelocally compact and complete. It follows from Lemma 1.2 that a metric space (X, d) isproper if and only if the map (x, y) 7−→ (d(x, y), y) from X ×X to R×X is proper.

Parts of following result are proven in [1, Thm. 3.1], in [16, 5.2–5.6] for separablespaces, and in [17, Thm. 2.3] 3.

Theorem 2.2. Let X be a proper metric space. Then the map IE(X) × X −→ X × Xthat maps (g, x) to (gx, x) is proper. The topological monoid IE(X) is locally compact andsecond countable. The subgroup Iso(X) is closed in IE(X) and Iso(X) × X −→ X is alocally compact, second countable, proper transformation group.

Proof. We first show that the map F : (g, x) 7−→ (d(gx, x), x) is proper. If C ⊆ R andB ⊆ X are compact, we claim that the set

AB,C = {g ∈ IE(X) | there is b ∈ B such that d(gb, b) ∈ C}3The proof given in [17] contains a gap; we fail to explain why Iso(X) ⊆ IE(X) is closed.

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is compact. We choose r > 0 with C ⊆ [−r, r], and with diam(B) ≤ r. For g ∈ AB,C andb ∈ B with d(gb, b) ∈ C and z ∈ X we have then

d(gz, z) ≤ d(gz, gb) + d(gb, b) + d(b, z) ≤ r + 2d(z, b) ≤ 3r + 2d(z, B).

Hence AB,C ⊆∏

z∈X B3r+2d(z,B)(z) ⊆∏

z∈X X is compact. It follows from Lemma 1.2 thatF is proper. Since F factors as (g, x) 7−→ (gx, x) 7−→ (d(gx, x), x), the map (g, x) 7−→(gx, x) is also proper by 1.1. In particular, IE(X)×X is locally compact and thus IE(X)is locally compact.

Every compact metrizable space is second countable. Being proper, X is therefore aunion of countably many open and second countable sets. Hence X is second countable,and therefore the compact-open topology on IE(X) is also second countable [14, XII.5.2].

Suppose that g ∈ IE(X) is contained in the closure of Iso(X). We fix p ∈ X andr > 0, such that d(p, gp) < r. The set K = {h ∈ IE(X) | d(p, hp) ≤ r} is compact andcontains g. We put L = K ∩ Iso(X) and we note that g ∈ L. We claim that idX ∈ gL.For every neighborhood V of g, there is an element h ∈ L ∩ V . But then also h−1 ∈ Land thus idX ∈ V L ⊆ V L. Since L is compact, Wallace’ Lemma [14, XI.2.6], [15, 3.2.10]shows that idX ∈ gL. In particular, there exists h ∈ IE(X) such that gh = idX . Thus gis surjective, and hence in Iso(X).

Since Iso(X) ⊆ IE(X) is closed, the restriction Iso(X)×X −→ X ×X is proper.

The following example shows that the assumption of X being proper is essential inthe previous theorem.

Example 2.3. Let X be a countably infinite set, with the discrete metric d(x, y) = 1whenever x 6= y. Then X is locally compact, but not proper. The monoid IE(X) consistsof all injective maps X −→ X , while the isometry group of X consists of all permutationsof X . For every g ∈ IE(X) and every finite subset E ⊆ X , there exists a permutation hof X that agrees with g on E. Thus Iso(X) is dense in IE(X). Neither IE(X) nor Iso(X)is locally compact. 4

Example 1.8 shows that a group can act isometrically and faithfully on a proper metricspace without acting properly. However, every group that acts faithfully, isometricallyand properly on a proper metric space carries necessarily the topology of pointwise con-vergence, as we show now.

Proposition 2.4. Let (X, d) be a proper metric space and let G×X −→ X be a properand isometric action. Then the associated homomorphism G −→ Iso(X) is continuousand proper. In particular, G is locally compact.

4Both spaces are Polish. The monoid IE(X) is the left Weil completion of Iso(X), while the groupIso(X) is Raikov complete.

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Proof. For every z ∈ X , the evaluation map G −→ X that maps g to gz is continuous.Hence the homomorphism G −→ Iso(X) is continuous for the topology of pointwiseconvergence on Iso(X). The claim follows now from Lemma 1.6, with K = Iso(X) andf = idX .

We recall that a group Γ acts properly discontinuously on a space X if the action isproper with respect to the discrete topology on Γ. 5

Corollary 2.5. Suppose that a group Γ acts faithfully and isometrically on a proper metricspace (X, d). Then the following are equivalent.(i) The action is properly discontinuous.(ii) The group Γ is discrete in the topology of pointwise convergence.(iii) The group Γ is discrete in the compact-open topology.

We note that condition (ii) may be stated as follows: there are points p1, . . . , pn ∈ Xand ε > 0 such that every g ∈ Γ with d(gpj, pj) < ε, for j = 1, . . . , n, is necessarily theidentity element.

3 Metric simplicial complexes

Now we turn to proper actions on metric simplicial complexes.

Definition 3.1. Let V be a set. A simplicial complex ∆ with vertex set V is a set offinite subsets of V that satisfies the following axioms.(i) If a ⊆ b ∈ ∆, then a ∈ ∆.(ii)

⋃∆ = V .

The geometric realization |∆| of ∆ is defined as follows [23, 3.1.14]. In the real vectorspace R(V ) spanned by the basis V , we let |a| denote the convex hull of the simplex a ∈ ∆.The geometric realization of ∆ is the set |∆| =

⋃{|a| | a ∈ ∆}. The subsets |a| ⊆ |∆| will

also be called simplices. The weak topology on |∆| is defined as follows. Each simplex |a|carries its natural compact topology as a subset of Ra, and a subset A ⊆ |∆| is closed ifA ∩ |a| is closed in |a| for all a ∈ ∆.

Lemma 3.2. Let ∆ be a simplicial complex. Then the vertex set V ⊆ |∆| is closed anddiscrete in the weak topology.

5Again, different authors define properly discontinuous actions in rather different ways. Kapovich’sarticle [18] is an excellent reference for a comparison of differing definitions. Some authors require inaddition that the action is free.

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Proof. The vertex set is closed from the definition of the weak topology. If v is a vertex,then

U = |∆| −⋃

{|a| | a ∈ ∆ and v 6∈ a}.is an open neighborhood of v which contains no vertex besides v.

Suppose that ∆′ is another simplicial complex, with vertex set V ′. A map ϕ : V −→ V ′

is called simplicial if it maps simplices to simplices. Then ϕ extends to a map ϕ : ∆ −→ ∆′,and to a linear map ϕ : |∆| −→ |∆′| which is continuous with respect to the weak topolo-gies.

Lemma 3.3. Let ∆ and ∆′ be simplicial complexes on vertex sets V, V ′. Then the fol-lowing topologies on the set S(∆,∆′) of all simplicial maps from ∆ to ∆′ coincide.(i) The compact-open topology, for the weak topologies on |∆| and |∆′|.(ii) The topology of pointwise convergence, as maps from |∆| to |∆′|, for any topology

on |∆′| in which the vertex set V ′ ⊆ |∆′| is discrete.(iii) The topology of pointwise convergence, as maps from V to the discrete space V ′.

Case (ii) in the lemma applies in particular to the weak topology on |∆′| by Lemma 3.2.

Proof. We first show that the topologies in (ii) and (iii) on S(∆,∆′) are equal. If ϕ, ψ ∈S(∆,∆′) agree on a finite subset V0 ⊆ V , then they agree on the finite subcomplex ∆0

spanned by the vertex set V0. Hence the topology in (ii) is coarser than the topology in(iii). On the other hand, if ϕ ∈ S(∆,∆′) and if V0 ⊆ V is finite, then we can choose forevery v ∈ V0 a neighborhood Uv of ϕ(v) that contains no vertex besides ϕ(v). Hence ifψ ∈ S(∆,∆′) and if ψ(v) ∈ Uv holds for all v ∈ V0, then ϕ and ψ agree on V0. This showsthat the topology in (iii) is coarser than the topology in (ii).

Now we compare the topologies in (i) and (iii). Suppose that A ⊆ |∆| is compact,that U ⊆ |∆′| is open, and that ϕ ∈ S(∆,∆′) is a simplicial map with ϕ(A) ⊆ U . SinceA is compact, there is a finite subcomplex ∆0 ⊆ ∆ with A ⊆ |∆0| [23, Ch. 3.1.19]. Let V0denote the (finite) vertex set of ∆0. If ψ is a simplicial map which agrees with ϕ on V0,then ψ|∆0

= ϕ|∆0. In particular, ψ(A) = ϕ(A) ⊆ U . This shows that the compact-open

topology in (i) is coarser than the topology in (iii).The topology in (iii) agrees with the topology of pointwise convergence in (ii), for the

weak topology on |∆′|. Since the topology of pointwise convergence is always coarser thanthe compact-open topology, all three topologies are equal.

The following is a basic example of a piecewise linear locally euclidean metrizablesimplicial complex which fails to be proper.

Example 3.4. PutV = { 1

n| for n = 1, 2, 3, 4, . . .} ⊆ (0, 1]

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and let ∆ be the 1-dimensional simplicial complex whose 1-simplices are the sets { 1n+1

, 1n},

for n = 1, 2, 3, 4, . . .. Then |∆| is in the weak topology homeomorphic to the half-openinterval (0, 1]. For the standard euclidean metric, |∆| is not complete and therefore not aproper metric space, even though ∆ is a locally finite simplicial complex.

We recall the definition of an Mκ-simplicial complex [7, I.7].

Definition 3.5. Suppose we are given a simplicial complex ∆ and a real number κ. Ageodesic simplex of constant sectional curvature κ is the convex hull of a finite set ofpoints in general position in the simply connected Riemannian manifold Mκ of constantsectional curvature κ [7, I.7.1]. AnMκ-structure on ∆ consists of a collection S of geodesicsimplices of constant sectional curvature κ. For each a ∈ ∆, there is a geodesic simplexsa ∈ S and an affine bijection

σa : |a| −→ sa

in the sense of [7, I.7A.7]. These bijections are subject to the usual compatibility condition:if b ⊆ a, then σa ◦ σ−1

b : sb −→ sa maps sb isometrically onto a face of sa [7, I.7A.9]. Inthis way, every simplex |a| carries a well-defined metric da of constant sectional curvatureκ. The associated intrinsic pseudometric on |∆| is defined as follows. An m-stringx = (x0, . . . , xm) is a finite sequence in |∆|, such that consecutive points xi−1, xi arecontained in a common simplex |ai|. The length of the string is

ℓ(x) =m∑

i=1

dai(xi−1, xi).

The intrinsic pseudo-metric distance d(x, y) between x, y ∈ |∆| is then the infimum ofthe lengths of all strings joining x and y. If S is finite, one says that the Mκ-structure on∆ has finitely many shapes. In this case (|∆|, d) is a complete geodesic metric space [7,I.7.19].

The simplicial complex in Example 3.4 is locally finite, but not with finitely manyshapes.

Lemma 3.6. Let ∆ be a simplicial complex with an Mκ-structure with finitely manyshapes. Then the intrinsic distance between distinct vertices is bounded away from 0, andhence V is closed and discrete in the metric space |∆|.

Proof. For a vertex v we put as in [7, Def. I.7.8]

ε(v) = inf{ε(v, a) | v ∈ a},

whereε(v, a) = inf{da(v, |b|) | b ⊆ a and v 6∈ b}.

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Since S is finite, ε(v) is bounded away from 0 as v varies in V . Now [7, Lem. I.7.9] saysthat if w is another vertex with d(v, w) < ε(v), then v = w.

Theorem 3.7. Let ∆ be a simplicial complex with an Mκ-structure with finitely manyshapes. Then the following are equivalent.(i) ∆ is locally finite.(ii) |∆| is a proper metric space for the intrinsic metric.

If ∆ is locally finite, then the weak topology and the metric topology on |∆| coincide.

Proof. For each a ∈ ∆, the map σa : |a| −→ sa is a homeomorphism, where |a| carriesthe metric topology coming from d [7, I.7.6]. In particular, each simplex |a| is compact.Since S is finite, the distance between two points x, y is realized as the infimum of thelengths of all taut strings 6 between x and y [7, I.7.20 and I.7.24].

Suppose that ∆ is locally finite, that x ∈ |∆|, and that r > 0. We claim that theclosed ball Br(x) is compact. For every r > 0 there is a number N > 0 such that everytaut m-string (x0, x1, . . . , xm) between two points x = x0, y = xm at distance d(x, y) ≤ rhas m ≤ N [7, I.7.28]. It follows that there is a sequence of simplices a1, . . . , am ∈ ∆, withxi−1, xi ∈ |ai|. Since ∆ is locally finite, the set of all such m-chains of simplices startingat x with m ≤ N is finite. Hence Br(x) is contained in a finite subcomplex ∆0, that is,Br(x) ⊆ |∆0|. Since all simplices |a| are compact, |∆0| is compact in the metric topology.

Suppose that ∆ is not locally finite. The vertex set V is closed and discrete byLemma 3.6. By assumption, there is a vertex v that is contained in infinitely manysimplices. Put

r = sup{d(v, w) | v, w ∈ V, v, w are in a common simplex}.

Since S is finite, r < ∞. Then the closed ball Br(v) contains the infinite closed discreteset V ∩ Br(v). In particular, Br(v) is not compact.

For the last claim we note that the metric topology on |∆| is coarser than the weaktopology. Let A ⊆ |∆| be compact in the metric topology. Then A is contained insome closed ball Br(v). The argument above shows that Br(v) is contained in a finitesubcomplex |∆0|. In particular, A is contained in |∆0|, and |∆0| is compact in the weaktopology. Since A is also closed in the weak topology, A is compact in the weak topology.Therefore the identity is a proper map from the weak topology to the metric topology,and hence a homeomorphism.

Proposition 3.8. Let ∆ be a simplicial complex with an Mκ-structure with finitely manyshapes, and let d denote the intrinsic metric. Then the subgroup Iso(|∆|) ∩ S(∆,∆)

6An m-string (x0, . . . , xm) in |∆| is taut if no three consecutive points xi−1, xi, xi+1 are in a commonsimplex, and if all triples xi−1, xi, xi+1 are ’straight’ in a weak sense.

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consisting of all simplicial isometries is a closed and totally disconnected subgroup ofIso(|∆|).

If ∆ is locally finite, then Iso(|∆|) ∩ S(∆,∆) is locally compact, totally disconnected,and second countable.

Proof. Let g be an isometry which is in the closure of Iso(|∆|) ∩ S(∆,∆), and let abe a simplex. For each vertex v ∈ a and every neighborhood U of g(v), there is someh ∈ Iso(|∆|) ∩ S(∆,∆) with h(v) ∈ U . Since V is closed and discrete by Lemma 3.6, thepoint g(v) is a vertex. Hence there exists h ∈ Iso(|∆|) ∩ S(∆,∆) with g(v) = h(v) for allv ∈ a. It follows that g||a| = h||a| and therefore g is simplicial.

Since the space V V is totally disconnected and since S(∆,∆) carries the subspacetopology by Lemma 3.3, Iso(|∆|)∩S(∆,∆) is totally disconnected. The last claim followsfrom Theorem 3.7 and Theorem 2.2.

4 Buildings

We recall the definition of a building as a chamber graph. The books [4, 8, 13, 22, 24, 25]are excellent references. Let I be a finite set and let (mi,j)i,j∈I be a symmetric matrixwith entries in {1, 2, 3, 4, 5, . . . ,∞}, with mi,i = 1 for all i and mi,j ≥ 2 for all i 6= j. Theassociated Coxeter system (W, I) consists of the Coxeter group W , with the presentation

W = 〈I | (ij)mi,j = 1 if mi,j 6= ∞〉,

and its generating set I. It follows that the product ij has order mi,j in W [22, Lem. 2.1].The length function of W with respect to the generating set I is denoted by ℓ. A pre-sentation of a group element w = i1 · · · im in terms of generators i1, . . . , im ∈ I is calledreduced or minimal if ℓ(w) = m.

Definition 4.1. Let (W, I) be a Coxeter system. A building B = (C,EC, t, δ) of type(W, I) consists of a simplicial graph (C,EC) with vertex set C, edge set EC, and twomaps

t : EC −→ I and δ : C × C −→ W.

The elements of C are called chambers, the map t is the edge coloring, and the map δ isthe W -valued distance function. Two chambers which are adjacent by an edge of color iare called i-adjacent. The graph (C,EC) is called the chamber graph. A gallery in thebuilding is a sequence of chambers (c0, . . . , cm), such that cs−1 is is-adjacent to cs, fors = 1, . . . , m. The type of the gallery is the string (i1, . . . , im) ∈ Im. The maps t and δare subject to the following axioms. 7

7We view the empty graph as the unique building corresponding to I = ∅.

12

(B1) For every chamber a ∈ C and i ∈ I, there is at least one chamber b which isi-adjacent to a. If another chamber c is i-adjacent to a, then b and c are alsoi-adjacent.

(B2) Suppose that w ∈ W and that w = i1 · · · im is a reduced presentation. Thenδ(a, b) = w holds for two chambers a, b if and only if there is a gallery of type(i1, . . . , im) from a to b.

It follows that δ(a, b) = 1 holds if and only if a = b, and that δ(b, a) = δ(a, b)−1.

The cardinality of the set I is called the rank of the building. We note that twochambers a, b are i-adjacent if and only if δ(a, b) = i. Hence δ determines EC and tuniquely. Conversely, δ is determined by EC and t by (B2). Buildings can be thought ofas generalizations of Cayley graphs of Coxeter groups.

Example 4.2. The most basic example of a building of type (W, I) is the Cayley graphof a Coxeter system (W, I). Thus C = W and EC = {{w,wi} | w ∈ W and i ∈ I}, withδ(a, b) = a−1b and t({w,wi}) = δ(w,wi) = i. This building has the particular propertythat for every chamber a and every i ∈ I, there is precisely one other chamber b whichis i-adjacent to a. Such buildings are called thin. One can show that every thin buildingof type (W, I) is isomorphic to the Cayley graph of (W, I). (For the proof one fixes achamber c and considers the map a 7−→ δ(c, a) from C to W .) A building is called thickif for every chamber a and every i ∈ I, there are at least two distinct chambers b, c whichare i-adjacent to a.

Example 4.3. Let (Gi)i∈I be a finite family of nontrivial groups Gi, and let

G =∐

i∈I

Gi

denote their coproduct. If we put C = G and if we call a, b ∈ G i-adjacent if 1 6= a−1b ∈Gi ⊆ G, then we obtain a building. The corresponding Coxeter system has mi,j = ∞ forall i 6= j. This building is thick if and only if every Gi has at least 3 elements.

The construction can be modified [13, 18.1.10]. Suppose that Γ is a simplicial graphon the vertex set I. We put

GΓ = G/〈〈[Gi, Gj] if i, j are adjacent in Γ〉〉.

This group is called the graph product of theGi. We put C = GΓ and we define i-adjacencyas before by 1 6= a−1b ∈ Gi ⊆ GΓ. The associated Coxeter system has mi,j = ∞ wheneveri and j are different and not joined by an edge in Γ, and mi,j = 2 if i and j are joined byan edge.

Buildings of this type are called right-angled buildings. These buildings are very differ-ent from the so-called affine or spherical buildings. The latter are, by Tits’ fundamental

13

classification results, closely related to semisimple algebraic groups [24, 25]. Right-angledbuildings, on the other hand, are similar to trees. They are easy to construct, while theirautomorphism groups have many interesting subgroups.

A subbuilding of a building is a full subgraph which is a building in its own right(of the same type (W, I)). A subbuilding which is thin is called an apartment in theambient building. One can show that any two chambers in a building are contained insome apartment [22, Cor. 3.7]. Every apartment is by Example 4.2 isomorphic to theCayley graph of (W, I). If A,A′ are two apartments, then there is a color-preservinggraph isomorphism A −→ A′ that fixes A ∩ A′ element-wise [22, Thm. 3.11].

Apartments can be viewed as coordinate charts in buildings, similarly to the coordinatecharts in a manifold. An atlas A is a set of apartments such that every pair of chambersof the building is contained in at least one apartment in A. Every building has a uniquemaximal atlas, consisting of all apartments in the building.

Definition 4.4. An automorphism h of a building is a graph automorphism that preservesthe coloring of the edges. Then the distance function δ is h-invariant, δ(ha, hb) = δ(a, b).Let G be a group that acts on a building B as a group of automorphisms. The action isWeyl-transitive if for all w ∈ W and all chambers a, b, a′, b′ with w = δ(a, b) = δ(a′, b′)there is some g ∈ G with ga = a′ and gb = b′. The action is strongly transitive with respectto an atlas A if the action preserves A, and if for every a ∈ A ∈ A and a′ ∈ A′ ∈ A thereis some g ∈ G with ga = a′ and gA = A′. Strongly transitive actions are in particularWeyl-transitive. The converse is in general not true [3].

In Example 4.2, the Coxeter group W acts (from the left) on itself, and this action isstrongly transitive and Weyl-transitive. In Example 4.3, the automorphism group of theright-angled building (which contains G resp. GΓ as a subgroup) acts strongly transitivelyon the maximal atlas [10, Cor. 1.2].

A gallery in a building from a chamber a to a chamber b can be viewed as a simplicialpath in the graph. One can show that ℓ(δ(a, b)) is the graph-theoretic distance between aand b, i.e. a gallery (c0, . . . , cm) of type (i1, . . . , im) is minimal if and only if w = i1 · · · imis reduced. Moreover, one can show that a minimal gallery is uniquely determined by itsendpoints and its type [22, 3.1]. Thus, a group G acts Weyl-transitively on a building ifand only if for every reduced type (i1, . . . , im) and every pair of galleries (c0, . . . , cm) and(c′0, . . . , c

′m) of the same type (i1, . . . , im) there is a g ∈ G with

(gc0, . . . , gcm) = (c′0, . . . , c′m).

Now we turn to infinite galleries.

Definition 4.5. We call an infinite string (is)s∈Z in IZ reduced if every finite substring(is, is+1, . . . , is+r) is reduced, i.e. if ℓ(isis+1 · · · is+r) = r + 1 for all s ∈ Z and r ≥ 0. One

14

can show that the Coxeter group W is infinite if and only if every finite reduced string(i1, . . . , im) is part of an infinite reduced string [22, Thm. 2.16]. An infinite gallery of type(is)s∈Z is a sequence of chambers (cs)s∈Z, where cs−1 is is-adjacent to cs, for all s ∈ Z.If W is infinite, then every minimal gallery is part of an infinite reduced gallery in thebuilding. Moreover, every infinite reduced gallery is contained in some apartment in thebuilding [22, Thm. 3.6].

We say that a group G acts infinitely Weyl-transitively on a building if for everyinfinite reduced string (is)s∈Z in IZ and every pair of infinite galleries (cs)s∈Z and (c′s)s∈Zof type (is)s∈Z there is an element g ∈ G with gcs = c′s, for all s ∈ Z.

Suppose that W is infinite and that A is the maximal atlas of a building B of type(W, I). For a group G acting by automorphisms on B, we have the implications

Strongly transitive on A =⇒ infinitely Weyl-transitive =⇒ Weyl-transitive.

IfW is finite, then every Weyl-transitive action is strongly transitive on the maximal atlas(and the maximal atlas is the unique atlas).

5 Proper actions on buildings

A building is called locally finite if for every color i ∈ I and every chamber a, the set ofall chambers which are i-adjacent to a is finite. In Example 4.3 this holds if and only ifthe groups Gi are finite.

Theorem 5.1. Let B be a building and put d(a, b) = ℓ(δ(a, b)). Then (C, d) is a completemetric space. With respect to the topology of pointwise convergence, the automorphismgroup Aut(B) is a closed and totally disconnected subgroup of Iso(C). If B is locally finite,then Aut(B) is locally compact and the action on C is proper.

We note that the topology of pointwise convergence on Aut(B) depends only on thediscrete space C, not on the structure of the building (or the specific metric d). SinceAut(X) is contained in the totally disconnected space CC , it is totally disconnected.

Proof. We have d(a, b) = n if and only if there is a minimal gallery of length n from ato b [22, 3.1]. Hence d is a (complete) metric on C. By Lemma 2.1, the isometry groupof (C, d) is a topological group in the topology of pointwise convergence. Suppose thatg ∈ Iso(C) is not an automorphism of B. Then there exists a pair of i-adjacent vertices a, bso that ga, gb are j-adjacent, for some j 6= i. The set {h ∈ Iso(C) | ha = ga and hb = gb}is a neighborhood of g and disjoint from Aut(B). Hence Aut(B) is a closed subgroup inIso(C).

15

If B is locally finite, then the graph (C,EC) is locally finite and therefore every metricball in C is finite and hence compact. The isometry group Iso(C) of (C, d) is thereforelocally compact and acts properly on C by Theorem 2.2. Thus Aut(B) is also locallycompact and acts properly on C by Corollary 1.7.

Theorem 5.2. Let B be a building of type (W, I), with W infinite, and let G be a groupthat acts properly as a group of automorphisms on B. Then the following are equivalent.(i) The action is infinitely Weyl-transitive.(ii) The action is Weyl-transitive.

Proof. As we observed in the last section, (i) implies (ii).Suppose that the action is Weyl-transitive, that (is)s∈Z is reduced and that (cs)s∈Z

and (c′s)s∈Z are two infinite galleries, both of type (is)s∈Z. For each s ≥ 0 put

Gs = {g ∈ G | g(cs) = c′s and g(c−s) = c′−s}.

The Gs are compact (because the action is proper) and nonempty (because the action isWeyl-transitive). Moreover, Gs ⊇ Gs+1. Hence

⋂{Gs | s ≥ 0} 6= ∅.

So far, we have considered buildings as edge-colored graphs. Now we turn to geometricrealizations. Let B be a building. The chamber graph (C,EC) is then a 1-dimensionalsimplicial complex, which we denote by ∆C .

Proposition 5.3. Let B be a building. Then the following topologies on Aut(B) coincide.(i) The topology of pointwise convergence on the set of chambers C.(ii) The topology of pointwise convergence, for any Aut(B)-invariant topology on |∆C |

in which C is a discrete subset.(iii) The compact-open topology, for the weak topology on |∆C |.Proof. This follows from Lemma 3.3.

Now we consider the classical Tits realization of a building as a simplicial complex.

Definition 5.4. Let (W, I) be a Coxeter system and let J ⊆ I be a subset. Let WJ ⊆Wdenote the subgroup generated by J . Then (WJ , J) is again a Coxeter system, for therestricted Coxeter matrix (mi,j)i,j∈J [22, Cor. 2.14]. Let c be a chamber in a buildingof type (W, I) and let J ⊆ I. The J-residue ResJ(c) of c is the subgraph consisting ofall chambers that can be reached from c using galleries whose edge colors are in J , andthe edges between these chambers. Then ResJ(c) is again a building of type (WJ , J) [22,Thm. 3.5].

The poset {ResJ(c) | c ∈ C and J ⊆ I}, ordered by reversed inclusion, is poset-isomorphic to a simplicial complex ∆T . The vertex set of ∆T is

V = {ResI−{i}(c) | c ∈ C and i ∈ I},

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The simplices are finite subsets

a = {ResI−{i0}(c),ResI−{i1}(c), . . . ,ResI−{ik}(c)},

with {i0, . . . , ik} ⊆ I, and a corresponds to ResI−{i0,...,ik}(c) in the poset isomorphism [4,5.6]. In particular, the maximal simplices in ∆T correspond to the chambers in C. Wecall the geometric realization |∆T | of ∆T the Tits realization. The simplicial complex ∆T

is locally finite if and only if the building B is locally finite.

The Solomon-Tits Theorem [4, Thm. 4.127] asserts that

|∆T | ≃ {∗}

is contractible (in the weak topology) if W is infinite. If W is finite, then it contains a(unique) element w0 of maximal length. Then |∆T | has the homotopy type of a wedge ofspheres,

|∆T | ≃∨

Q

SSn−1,

where n = #I and Q = {a ∈ C | δ(a, c) = w0}, for some (any) c ∈ C.Earlier in this section, we endowed the automorphism group of a building with the

topology of pointwise convergence on the chamber set C. This topology works well withthe Tits realization.

Proposition 5.5. Let B be a building. Then the following topologies on Aut(B) coincide.(i) The topology of pointwise convergence on the set of chambers C.(ii) The topology of pointwise convergence on the set of vertices of ∆T .(iii) The topology of pointwise convergence on the Tits realization |∆T |, for any Aut(B)-

invariant topology on |∆T | in which the vertex set is discrete.(iv) The compact-open topology, for the weak topology on the Tits realization |∆T |.

Proof. The topologies (ii), (iii) and (iv) coincide by Lemma 3.3. If two automorphismsagree on a finite set of chambers, then they agree on the finite set of vertices of thecorresponding maximal simplices, and vice versa. Hence the topologies in (i) and (ii)coincide.

From Proposition 5.5 and Lemma 2.1 we obtain the following.

Corollary 5.6. Let B be a building and let d be a metric on the Tits realization |∆T |.If the metric is Aut(B)-invariant and if the distance between distinct vertices is boundedaway from 0, then the compact-open topology on Aut(B) with respect to the metric space(|∆T |, d) coincides with the topology of pointwise convergence on the chamber set C of thebuilding.

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The previous corollary is mainly interesting if W is finite (spherical) or of irreducibleeuclidean type, because then |∆T | carries an interesting invariant CAT(1) resp. CAT(0)metric. For many questions arising in geometric group theory, the Tits realization is notthe right geometric object. The Davis realization of a building B is defined as follows.

Definition 5.7. A residue ResJ(c) is called spherical if WJ is finite. Let V denote theset of all spherical residues of B. Let ∆D denote the simplicial complex whose simplicesare ascending chains ResJ0(c) ⊆ ResJ1(c) ⊆ · · · ⊆ ResJm(c), with J0 ⊆ J1 ⊆ · · · ⊆ Jm andwith WJm finite. Again, ∆D is locally finite if and only if B is locally finite. The Davisrealization of B is defined to be the set |∆D|. 8 The Davis realization is contractible (withrespect to the weak topology). More importantly, it admits a CAT(0) metric. This metricd is the intrinsic metric of an M0-structure on ∆D with finitely many shapes [13, 18.3]. Ifthe building is locally finite, then the Davis realization (|∆D|, d) is a proper metric spaceby Theorem 3.7, and Aut(B) is locally compact, totally disconnected, and acts properlyand isometrically on |∆D|.Proposition 5.8. Let B be a building. Then the following topologies on Aut(B) coincide.(i) The topology of pointwise convergence on the set of chambers C.(ii) The topology of pointwise convergence on the set of vertices of ∆D.(iii) The topology of pointwise convergence on the Davis realization |∆D|, for any Aut(B)-

invariant topology on |∆D| in which the vertex set is discrete.(iv) The compact-open topology, for the weak topology on the Davis realization |∆D|.Proof. The topologies (ii), (iii) and (iv) coincide by Lemma 3.3. Every chamber is avertex in ∆D, hence the topologies in (i) and (ii) coincide.

The following corollary applies in particular to the intrinsic CAT(0) metric on theDavis realization.

Corollary 5.9. Let B be a building and let d be a metric on the Davis realization |∆D|.If the metric is Aut(B)-invariant and if the distance between distinct vertices is boundedaway from 0, then the compact-open topology on Aut(B) with respect to the metric space(|∆D|, d) coincides with the topology of pointwise convergence on the chamber set C of thebuilding.

We end with a geometric variant of Theorem 5.2. By an infinite line L in a metricspace X we mean an isometric embedding L : R −→ X . Suppose that |∆D| is the Davisrealization of a building B, with its intrinsic CAT(0) metric d. Then for every infinite lineL in |∆D|, there is an apartment A in B such that

L(R) ⊆ |AD| ⊆ |∆D|,8The simplicial complex ∆D is not quite the Davis complex ; rather, it is the barycentric subdivision

of the Davis complex. However, |∆D| is the geometric realization of the Davis complex.

18

where |AD| is the Davis realization of A [12, Thm. E]. Lines exist in the Davis complex,provided that W is infinite.

We say that two infinite lines L, L′ in |∆D| are of the same type if there exist apartmentsA,A′ in B, with L(R) ⊆ |AD| and L′(R) ⊆ |A′

D|, and a building isomorphism ψ : A −→ A′

with L′ = ψ ◦ L.

Proposition 5.10. Let B be a building of type (W, I), with W infinite, and let G bea group that acts properly as a group of automorphisms on B. If the action is Weyl-transitive, then it is transitive on the set of all infinite lines of any given type.

Proof. Let L, L′ be two infinite lines of the same type, and let A,A′ be apartments inB whose Davis realizations contain L and L′, respectively. Let ψ : A −→ A′ be anisomorphism with L′ = ψ ◦ L. For every t ∈ R we choose a maximal simplex st in AD

with L(t) ∈ |st|. Since st is maximal, it is of the form

st = {{ct} ⊆ Resj(ct) ⊆ · · · ⊆ ResJmt(ct)},

for some chamber ct and some maximal spherical subset Jmt⊆ I. We put c′t = ψ(ct) and

s′t = ψ(st). For every t ≥ 0 there is some g ∈ G with

g(ct) = c′t and g(c−t) = c′−t,

because the action of G is Weyl-transitive. Then

g(L(t)) = L′(t) and g(L(−t)) = L′(−t),

because g(st) = s′t and g(s−t) = s′−t. We put

Gt = {g ∈ G | g(L(t)) = L′(t) and g(L(−t)) = L′(−t)}.

Then Gt is compact (because the action is proper) and nonempty by the remark above.For g ∈ Gt and −t ≤ s ≤ t we have g(L(s)) = L′(s), because geodesics in CAT(0) spacesare unique. Hence Gr+t ⊆ Gt for r ≥ 0. Thus

⋂{Gt | t ≥ 0} 6= ∅.

I do not know if transitivity on the set of infinite lines of any fixed type implies Weyl-transitivity. This is related to the question which geodesics in the Davis realization arecontained in infinite lines.

We finally apply our results to euclidean buildings and we recall the relevant notions.

Definition 5.11. Suppose that H is a collection of affine hyperplanes in some euclideanspace Rm, such that H is locally finite and invariant under the group W generated by thereflections along the hyperplanes in H. The hyperplanes in H are also called walls. If theaction ofW on Rm is cocompact, then there is a compact convex polytope C, bounded by

19

a finite set of hyperplanes from H, which is a fundamental domain for the W -action. Thereflections i1, . . . , ir along the codimension 1 faces of C generate W as a Coxeter group,and (W, I) is called a Coxeter system of euclidean type, for I = {i1, . . . , ir}. We refer to[8, Ch. VI.1].

The Davis realization of the Coxeter system in its CAT(0) metric is then isometric tothe euclidean space Rm. If the Coxeter system is irreducible, then the Tits realizationcoincides with the Davis realization, and C is a simplex. Otherwise, the Davis realizationis the cartesian product of the Tits (or Davis) realizations of the irreducible componentsof the Coxeter system, and C is a cartesian product of simplices (a complex polysimplicialin the language of [9]). We call a building B of type (W, I) a euclidean building.

The books [4], [8], [22] and [25] discuss mainly the case of euclidean buildings where theCoxeter system is irreducible. However, for the results that we consider here irreducibilityis not really important. One just has to keep in mind that the Davis realization isthe cartesian product of the Davis realizations of the irreducible factors, and that thefundamental chamber is no longer a simplex, but rather a cartesian product of simplices.Examples of euclidean buildings arise from semisimple algebraic groups over fields withdiscrete valuations [9]. The CAT(0) Davis realization of a euclidean building is a euclideanbuilding in the sense of Kleiner and Leeb [19] (but not conversely). We refer to [21], [13],and [4, Ch. 12].

Theorem 5.12. Let B be a euclidean building and let G be a group that acts properly asa group of automorphisms on B. Then the following are equivalent.(i) The G-action is Weyl-transitive.(ii) The G-action is strongly transitive on the maximal atlas.

Proof. We noted before that (ii) implies (i). Suppose that the action is Weyl-transitive.By Proposition 5.10, it is transitive on all infinite lines of a fixed type in its Davis realiza-tion. Let A be an apartment in B. If the infinite line L in |AD| is not at bounded distancefrom any wall in |AD|, then L is a regular geodesic in the sense of [19], and A is the uniqueapartment containing L [19, 4.6.4]. It follows that G acts transitively on the set of allapartments in B, and hence transitively on the maximal atlas. The Coxeter groupW actssharply transitively on the chambers in A. For every w ∈ W , the composite L′ = w ◦ Lis an infinite line of the same type as L. It follows that there is an element g ∈ G withgL′ = L. This element fixes A (because A is the unique apartment containing L and L′),and g|A = w because the only element of W that fixes L pointwise is the identity. Thisshows that the G-stabilizer of A acts transitively on the chambers in A, and hence G actsstrongly transitively on the maximal atlas of B.

Definition 5.13. Suppose that (W, I) is a Coxeter system of euclidean type acting onRm, as in Definition 5.11. The Coxeter group W decomposes as a semidirect product

20

W = Zm ⋊W0, for some spherical (i.e. finite) Coxeter group W0 ⊆ W . We may assumethat the origin 0 is the unique fixed point of W0. The walls passing through 0 subdivideRm into finitely many infinite cones which are called sectors or Weyl chambers.

We call an atlas A for a euclidean building B good [4, 11.8.4] if it satisfies the follow-ing extra condition: If S1 ⊆ A1 and S2 ⊆ A2 are sectors in apartments A1, A2 ∈ A, thenthere is an apartment A ∈ A and sectors S ′

1, S′2 ⊆ A, with S ′

1 ⊆ S1 and S ′2 ⊆ S2. In

this situation, there is a spherical building ∂AB attached to (B,A), the spherical buildingat infinity [4, 11.8.4] [25, Ch. 8]. The chambers of this spherical building correspond toequivalence classes of sectors in apartments in A, where two sectors are called equivalentif their intersection contains a sector. The apartments of ∂AB are in one-to-one corre-spondence with the apartments in A [25, Prop. 8.27]. The maximal atlas of B is a goodatlas [25, 7.24], and in this case the spherical building at infinity can be identified withthe Tits boundary of the Davis realization in its CAT(0) metric.

We apply Theorem 5.12 to Bruhat-Tits buildings, as defined in [25, Def. 13.1].

Definition 5.14. A Bruhat-Tits building (a Bruhat-Tits pair in the terminology of [25])consists of an irreducible thick euclidean building B and a good atlasA, such the sphericalbuilding at infinity ∂AB is a Moufang building (as defined in [25, 29.15]). Note that fromthe definition of a Moufang building, the dimension of B is at least 2. The classificationof these buildings is carried out in detail in [25]. We remark that every thick irreducibleeuclidean building of rank at least 4 (simplicial dimension at least 3) is automatically aBruhat-Tits building, for any good atlas [25, p. 119]. We refer also to the classification[17], and to [11] and [20] for conditions implying strong transitivity.

Theorem 5.15. Let (B,A) be a locally finite Bruhat-Tits building, and let G† ⊆ Aut(B)denote the group generated by all root groups of the spherical building ∂AB. Then theclosure H of G† (in the topology of pointwise convergence) is locally compact, secondcountable, totally disconnected, and acts properly and strongly transitively on the maximalatlas of B.

The following example may be instructive.

Example 5.16. Let p be a prime number, let

ν : Q −→ Z ∪ {∞}

denote the p-adic valuation and let

A = {a ∈ Q | ν(a) ≥ 0} ⊆ Q

denote the corresponding valuation ring. Then Qm is an A-module. A lattice L ⊆ Qm isan A-submodule which is free of rank m. Two such lattices L, L′ ⊆ Q are equivalent if

21

there is q ∈ Q× with qL = L′. Let B denote the euclidean building whose vertices are theequivalence classes of lattices [8, V.8]. The group G = PSLm(Q) acts strongly transitivelyon B, with respect to a good atlas A whose apartments correspond to m-tuples of pointsin general position in the projective space QPm−1. If m ≥ 3, then (B,A) is a Bruhat-Titsbuilding in the sense of Definition 5.14. The building B is locally finite and hence proper,but the group G = G† is not locally compact. However, it acts Weyl-transitively on B.The closure of G in the automorphism group is the group PSLm(Qp).

Proof of Theorem 5.15. By [25, Thm. 12.31], the group G† can be considered a subgroupof Aut(B). Since the building at infinity ∂AB is Moufang, the group G† acts transitivelyon the set of its apartments. By [25, 8.27], the group G† acts transitively on the atlas A.Let A be an apartment of B and let W ′ denote the subgroup of Aut(A) induced by thestabilizer of A in G†. By [25, Prop. 13.5 and Prop. 13.28], W ′ contains the Weyl groupW of A. In particular, W ′ acts transitively on the chambers of A. Thus G† acts stronglytransitively on A. In particular, G† acts Weyl-transitively on B.

Since Aut(B) is closed in the isometry group Iso(|∆D|) of the Davis realization byTheorem 5.1, the group H is contained in Aut(B). Moreover, the group H acts properlyon B. Hence H acts strongly transitively on the maximal atlas by Theorem 5.12.

The group Iso(|∆D|) is locally compact and second countable by Theorem 2.2, andAut(B) is totally disconnected by Theorem 5.1.

Acknowledgments

Arthur Bartels’ request for a solid reference for Theorem 5.15 initiated a first, 3-page ver-sion of the present article. I thank Bertrand Remy and Guy Rousseau for some remarks onBruhat-Tits buildings. Stefan Witzel, Herbert Abels and Polychronis Strantzalos spottedmistakes and pointed out some more references. In particular, the authors kindly informedme that the forthcoming book [2] will contain the results that we present in Section 2.Richard Weiss made helpful comments on the proof of Theorem 5.15 and the materialsurrounding it. Philip Moller read the article, spotted several mistakes and made goodsuggestions.

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