Buildings have finite asymptotic dimension

Jan Dymara ^{1} and Thomas Schick ^{2}

The goal of this note is to prove the following theorem, generalizing some results of Matsnev [Theorem 3.22, Mat].

Theorem 1.

The asymptotic dimension of any building is finite and equal to the asymptotic dimension of an apartment in that building.

Generally we use definitions and notation as in [D]. In particular, (W, S) is a finitely generated Coxeter system, C is a building with Weyl group W, |C| is the Davis realization of C. We will, however, confuse the Coxeter group and its abstract Coxeter complex, denoting both by W; in particular, |W | denotes the Davis complex. The W -valued distance in C will be denoted δ_{C}, while δ will be the gallery distance (i.e., δ = ℓ ∘ δ_{C}, where ℓ(w) is the shortest length of a word in generators S representing w). Basic properties of minimal galleries in buildings can be found in [R], [G]. We fix a chamber B ∈ C and define the B-based folding map as π: C → W, π(c) = δ_{C}(B, c). We also use π for the geometric realization |C| → |W | of this map. The word ‘building’ in the statement of Theorem 1 can be understood either as the discrete metric space (C, δ), or as the CAT (0) metric space (|C|, d) (these spaces are quasi–isometric). Neither thickness nor local finiteness of C is assumed.

Recall that a metric space X has asymptotic dimension ≤ n if for any d > 0 there exist n + 1 families U^{0}, …, U^{n} of subsets of X such that: (1) ⋃_{i} U^{i} is a uniformly bounded cover of X, and (2) for every i any two sets U, U′ ∈ U^{i} are d-disjoint: d(U, U′) := inf{d(x, y) | x ∈ U, y ∈ U′} ≥ d. Basic properties of asymptotic dimension can be found in [BD]. We only need the definition and the fact that Coxeter groups have finite asymptotic dimension (cf. [Theorem B, DJ]).

Now we start the proof of Theorem 1. Let n be the asymptotic dimension of the apartment W of C. Because W embeds isometrically into C, the asymptotic dimension of C is at least n. We have to prove that its asymptotic dimension is ≤ n.

Fix therefore d > 0. Let U^{0}, …, U^{n} be uniformly bounded families of 2d-disjoint sets in |W | such that ⋃_{i} U^{i} is a cover of |W |. For U ∈ U^{i} let N_{d}(U) = {x ∈ |W | | d(x, U) < d}, and let conv N_{d}(U) be the convex hull of this set in the CAT (0) space |W |. Note that diam(conv N_{d}(U)) ≤ diam(U) + 2d. Let C_{U} be the set of path–connected components of π^{−1}(conv N_{d}(U)). Put V_{U} = {π^{−1}(U) ∩ A | A ∈ C_{U}} and V^{i} = ⋃_{U∈U^{i}} V_{U}. It is obvious that ⋃_{i} V^{i} is a cover of |C|.

Claim: V^{i} are uniformly bounded families of d-disjoint sets.

Evidently, the claim implies Theorem 1. To prove the claim, we first establish d-disjointness. Let x ∈ V ∈ V_{U}, x′ ∈ V′ ∈ V_{U′} with V ≠ V′ and U, U′ ∈ U^{i}. There are two cases: U ≠ U′ and U = U′. In the first case d(x, x′) ≥ d(π(x), π(x′)). But π(x) ∈ U, π(x′) ∈ U′ and d(U, U′) ≥ 2d. Therefore d(V, V′) ≥ 2d > d. For the second case, assume that U = U′. Suppose also that V = π^{−1}(U) ∩ A and V′ = π^{−1}(U) ∩ A′, for some A, A′ ∈ C_{U}. Then the geodesic segment [x, x′] is not entirely contained in π^{−1}(conv N_{d}(U)) (otherwise A and A′, hence V and V′, coincide). Let p ∈ [x, x′] \ π^{−1}(conv N_{d}(U)); we have d(x, x′) = d(x, p) + d(p, x′) ≥ d(π(x), π(p)) + d(π(p), π(x′)) ≥ d + d ≥ d.

It remains to check uniform boundedness. Let V ∈ V_{U}; then V ⊆ A for some path–connected component A of π^{−1}(conv N_{d}(U)). It is enough to find a uniform bound on the diameter of A. We would like to make A gallery–connected. Since path–connected sets are usually not gallery–connected, we perform an auxiliary thickening construction. Let X be a subset of |C| or of |W |. We put T (X) = ⋃{Res(p) | p ∈ X}, where Res(p) = ⋃{|c| | p ∈ |c|}. Observe that Res(p) is a geometric realization of a spherical building, and

1 Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, dymara@math.uni.wroc.pl. Partially supported by MNiSW grant N201 012 32/0718 and by DFG grant 436 POL 113/110/0-1.

2 Mathem. Institut, Georg-August-Universität Göttingen, Bunsenstr. 3, 37073 Göttingen, Germany, schick@uni-math.gwdg.de. Partially supported by DFG grant 436 POL 113/110/0-1. spherical buildings are gallery–connected; therefore, any two points in A belong to chambers which can be connected by a gallery in T (A). Since T (A) ⊆ π^{−1}(T (conv N_{d}(U))), the set A is contained in a gallery– connected component of π^{−1}(T (conv N_{d}(U))). Observe that, uniformly in U, the diameter of T (conv N_{d}(U)) is bounded by R + 2d + 2κ, where R is the uniform bound on diameters of elements of the families U^{i}, and κ is the diameter of the realization of a chamber. Since the distances d and δ are quasi–isometric, it remains to prove the following lemma.

Lemma 1.

For any N > 0 there exists M > 0 such that if U is a subset of W of δ-diameter ≤ N, then any gallery–connected component V of π^{−1}(U) has δ-diameter ≤ M.

Lemma 1 follows form the next two lemmas.

Lemma 2.

Let W be a Coxeter group. For any R > 0 there exists D = D(R) such that for any subset U ⊂ |W | of diameter R satisfying d(|1|, U) > D there exists a codimension-one face of |1| such that the wall containing that face separates |1| from U.

Proof. For s ∈ S, let M_{s} be the wall containing the s-face of |1|, and let M_{s}^{−} be the open half-space with boundary M_{s} that does not intersect |1|. Let r be greater than twice the diameter of a chamber, and let b be the barycentre of |1|.

We denote with B_{r}(b) the open distance ball around b. We claim that {M_{s}^{−} \ B_{r}(b) | s ∈ S} is an open cover of |W | \ B_{r}(b). If a point q ∈ |W | \ B_{r}(b) is an interior point of a chamber, then we can consider a minimal gallery from 1 to that chamber; this gallery starts by crossing some wall M_{s}, and then q ∈ M_{s}^{−}. In general, we apply the above argument to the chamber c in Res(q) that is δ-closest to |1|. Any other chamber c′ in Res(q) can be connected to 1 by a minimal gallery passing through c, so that if a wall M_{s} separates 1 from c then it also separates 1 from c′. Therefore M_{s} separates b from all points in the interior of Res(q), in particular from q.

Intersecting open sets from the family {M_{s}^{−} \ B_{r}(b) | s ∈ S} with the distance sphere S_{r}(b), we obtain an open cover of this compact metric space. Let ϵ be the Lebesgue number of this cover. Put D = Rr/ϵ. We claim that the Lebesgue number of the cover {M_{s}^{−} \ B_{D}(b) | s ∈ S} of |W | \ B_{D}(b) is at least R. Indeed, let U′ be a subset of |W | \ B_{D}(b) of diameter ≤ R. Because W is a CAT (0)-space, the set U of intersection points of S_{r}(b) and geodesic intervals from b to points in U′ has diameter ≤ ϵ. Thus U ⊂ M_{s}^{−} for some s ∈ S. Now if a point q ∈ U′ did not belong to M_{s}^{−}, then S_{r}(b) ∩ [b, q] would not be in M_{s}^{−} either (because b, q belong to the convex set |W | \ M_{s}^{−}), contradicting U ⊂ M_{s}^{−}. QED(Lemma 2)

For X ⊆ C or X ⊆ W we denote by T (X) the union of all spherical residues that intersect X. Let U be a subset of W of δ-diameter ≤ N. There exists R > 0 depending only on N (and W) such that the d-diameter of |T (U)| is ≤ R. Iterated application of Lemma 2 provides a minimal gallery γ = (1, w_{1}, …, w_{k}) such that the wall between w_{i} and w_{i+1} separates w_{i} from T (U) and d(|w_{k}|, |T (U)|) ≤ D. Note that this separation property implies that every chamber which meets U can be joint to |1| by a minimal gallery which is a concatenation γδ, i.e. extending γ.

Lemma 3.

Let U and γ be given as just described. Recall that B is the “base” chamber in C, with π(B) = |1|. For any chamber c meeting π^{−1}(U) there is a minimal gallery from B to c whose π-projection extends γ.

For any gallery–connected component V ⊂ C of π^{−1}(U) there exists a chamber e ∈ π^{−1}(w_{k}) such that any minimal gallery from B to a chamber in V whose π-projection prolongs γ passes through e.

Proof. For each chamber c ∈ T (V) let P (c) be the set of all minimal galleries from b to c that are of the form ΓΔ, where π(Γ) = γ. We first show that this set is not empty. Choose an arbitrary minimal gallery from B to c with π-projection α, and a minimal gallery of the form γδ from |1| to π(c). α and γδ are minimal galleries with the same extremities, therefore are related by a sequence of Tits moves (cf. [Chap 4, Proposition 5, Bourb], or [Theorem 2.11, Ron]). This sequence lifts to a sequence of moves relating the original minimal gallery to a minimal ΓΔ with π-projection γδ, as required.

Next we claim that if Γ_{1}Δ_{1}, Γ_{2}Δ_{2} ∈ P (c), then Γ_{1} = Γ_{2}. In fact, π(Δ_{1}) and π(Δ_{2}) are minimal galleries in W with the same extremities, therefore again are related by a sequence of Tits moves. As above, this sequence lifts to a sequence of moves relating Δ_{1} and Δ′_{2} (and keeping extremities fixed), where Δ′_{2} has the same π-projection as Δ_{2}. Now Γ_{1}Δ′_{2} and Γ_{2}Δ_{2} have the same extremities and the same π-projections, so that they coincide.

Therefore we can define e(c) as Γ ∩ π^{−1}(w_{k}) for some ΓΔ ∈ P (c), and then e(c) does not depend on the choice of ΓΔ. We will now show that, for c ∈ V, e(c) does not depend on c. Since V is gallery connected, it is enough to check this independence for adjacent c, c′ ∈ V. There are three cases:

1) δ(w_{k}, π(c)) = δ(w_{k}, π(c′)). Then there is a chamber c′′ ∈ T (V) adjacent to both c and c′ whose πprojection is closer to w_{k} than that of c. And for any ΓΔ ∈ P (c′′) we have ΓΔc ∈ P (c), ΓΔc′ ∈ P (c′).

2) δ(w_{k}, π(c)) = δ(w_{k}, π(c′)) + 1. Then for any ΓΔ ∈ P (c) we have ΓΔc′ ∈ P (c′).

3) δ(w_{k}, π(c)) = δ(w_{k}, π(c′)) − 1. This is symmetric to 2), we just switch c and c′.

Finally, e = e(c), where c ∈ V, does not depend on c and is as claimed. QED(Lemma 3)

Let L is the maximal gallery distance between chambers of d-distance ≤ D + κ, where κ is the diameter of a realization of a chamber. As a result of Lemma 3, every chamber in V is at δ-distance < L + N from e, hence V has δ-diameter < 2L + 2N. This proves Lemma 1 and therefore Theorem 1.

The result leaves as an open task to determine the precise asymptotic dimension of a Coxeter complex W as above. These are particularly nice CAT (0)-complexes. In this context, it is plausible to expect that the asymptotic dimension coincides with the classical (microscopic) dimension for a suitable choice or modification of the complex W.

This coincidence has been established in model examples (coming from slightly different contexts), in particular for many homogeneous manifolds in [Corollary 3.6, CG] and in [Theorem 12, BD2]. For simply connected Riemannian manifolds with curvature bounded by c < 0, the result has been proved by Grave [Theorem 6.20, Gra]. For spaces with cocompact action of the isometry group which are Gromov hyperbolic, a related result has been obtained by Buyalo and Lebedeva [BL]. It is not known to us whether these results extend to non-positive curvature, or whether one can find counterexamples in this wider class. However, observe that it is not even known that the asymptotic dimension of every finite dimensional CAT (0)-space is finite.

Nonetheless, we conjecture that the answer to the following question is yes, at least in many good cases. Given this belief, it would be even more interesting to find counterexamples.

Question: Is it true that for the Coxeter complex W as considered in this note, the asymptotic dimension and the virtual cohomological dimension, i.e. the dimension of its Bestvina complex [Best] coincide? Note that the latter is also given by the dimension of the CAT (0)-boundary (compare [BM]).

It follows from [Corollary 4.11, Dr] that the asymptotic dimension of W is not smaller than its virtual cohomological dimension.

Acknowledgments.

We thank Alexander Dranishnikov and Jacek Świ ‘ atkowski for useful comments. This research was supported by German science foundation grant 436 POL 113/110/0-1, whose support for a visit of the first author to Georg-August-Universität Göttingen, and of the second author to Banach conference center, Bȩdlewo, is gratefully acknowledged. Moreover, we thank these institutions for their hospitality.

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