The ℓ2–homology of even Coxeter groups

Schroeder, Timothy A

doi:10.2140/agt.2009.9.1089

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

Timothy A Schroeder

Algebraic & Geometric Topology 9 (2009) 1089–1104

DOI: 10.2140/agt.2009.9.1089

Abstract

Given a Coxeter system $(W, S)$ , there is an associated CW–complex, denoted $Σ (W, S)$ (or simply $Σ$ ), on which $W$ acts properly and cocompactly. This is the Davis complex. The nerve $L$ of $(W, S)$ is a finite simplicial complex. When $L$ is a triangulation of $S^{3}$ , $Σ$ is a contractible $4$ –manifold. We prove that when $(W, S)$ is an even Coxeter system and $L$ is a flag triangulation of $S^{3}$ , then the reduced $ℓ^{2}$ –homology of $Σ$ 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

References

Publication

Received: 28 August 2008

Revised: 22 April 2009

Accepted: 22 April 2009

Published: 26 May 2009

Authors

Timothy A Schroeder
Department of Mathematics and Statistics Murray State University Murray, KY 42071 USA

Volume 9, issue 2 (2009)