#### Volume 9, issue 2 (2009)

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

### Timothy A Schroeder

Algebraic & Geometric Topology 9 (2009) 1089–1104
##### 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.

##### Keywords
Coxeter group, $\ell ^2$-homology, Singer Conjecture, Davis complex, aspherical manifold
##### Mathematical Subject Classification 2000
Primary: 20F55
Secondary: 57S30, 20J05, 57T15, 58H10