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  • Cited by 2
  • Print publication year: 1993
  • Online publication date: March 2010

2 - The Virtual Cohomological Dimension of Coxeter Groups


Abstract. For every finitely generated Coxeter group Γ we construct an acyclic complex of dimension vcd Γ where Γ acts cocompactly as a reflection group with finite stabilizers. This provides an effective calculation of vcdΓ in terms of the Coxeter diagram of Γ.


A Coxeter system is a pair (Γ, V) where Γ is a group (called a Coxeter group) and V is a finite set of generators for Γ all of which have order two, such that all relations in Γ are consequences of relations of the form (vw)m(V, W) = 1 for v, wV and m(u, w) denotes the order of vw in Γ. In particular, m(v, v) = 1 and m(v, w) = m(w, v) ∈ (2, 3,…, ∞).

In this note we address the question (see [Pr, problem 1]): What is the virtual cohomological dimension (vcd) of a Coxeter group?

Every Coxeter group can be realized as a group of matrices (see [Bou]), and consequently has a torsion free subgroup of finite index (by Selberg's lemma). Davis [Dav] has constructed a finite dimensional contractible complex (we review the construction below) where a given Coxeter group acts properly discontinuously, so that any torsion free subgroup acts freely. It follows from these remarks that the vcd of any Coxeter group is finite.

All finite Coxeter groups have been classified (see [Bou]), thus providing the answer to the above question when vcd equals zero. Another special case (when vcd ≤ 1) was resolved in [Pr-St].