The $\ell^2$–homology of even Coxeter groups

Abstract

Given a Coxeter system $\left(W,S\right)$, there is an associated CW–complex, denoted $\Sigma \left(W,S\right)$ (or simply $\Sigma$), on which $W$ acts properly and cocompactly. This is the Davis complex. The nerve $L$ of $\left(W,S\right)$ is a finite simplicial complex. When $L$ is a triangulation of ${\mathbb{S}}^3$, $\Sigma$ is a contractible $4$–manifold. We prove that when $\left(W,S\right)$ is an even Coxeter system and $L$ is a flag triangulation of ${\mathbb{S}}^3$, then the reduced ${\ell }^2$–homology of $\Sigma$ vanishes in all but the middle dimension.