finite asymptotic dimension

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 δC , while δ will be the gallery distance (i.e., δ = ` ◦ δC , where `(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) = δC(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, . . . ,U 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]).