Outline

For a finitely generated Coxeter group Γ, its virtual cohomological dimension over a (non-zero, associative) ring R, denoted vcdRΓ, is finite and has been described [8,1,11,13]. In [8], M. Davis introduced a contractible Γ-simplicial complex with finite stabilisers. The dimension of such a complex gives an upper bound for vcdRΓ. In [1], M. Bestvina gave an algorithm for constructing an R-acyclic Γ-simplicial complex with finite stabilisers of dimension exactly vcdRΓ, for R the integers or a prime field; he used this to exhibit a group whose cohomological dimension over the integers is finite but strictly greater than its cohomological dimension over the rationals. For the same rings, and for right-angled Coxeter groups, J. Harlander and H. Meinert [13] have shown that vcdHr is determined by the local structure of Davis’ complex and that Davis’ construction can be generalised to graph products of finite groups.

Our contribution splits into three parts. Firstly, Davis’ complex may be defined for infinitely generated Coxeter groups (and infinite graph products of finite groups). We determine which such groups Γ have finite virtual cohomological dimension over the integers, and give partial information concerning vcdzΓ. We discuss a form of Poincaré duality for simplicial complexes that are like manifolds from the point of view of R-homology, and give conditions for a (finite-index subgroup of a) Coxeter group to be a Poincare duality group over R.