Volume 5, issue 1 (2001)

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Vanishing theorems and conjectures for the $\ell^2$–homology of right-angled Coxeter groups

Michael W Davis and Boris Okun

Geometry & Topology 5 (2001) 7–74
 arXiv: math.GR/0102104
Abstract

Associated to any finite flag complex $L$ there is a right-angled Coxeter group ${W}_{L}$ and a cubical complex ${\Sigma }_{L}$ on which ${W}_{L}$ acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of ${\Sigma }_{L}$ is $L$ and (2) ${\Sigma }_{L}$ is contractible. It follows that if $L$ is a triangulation of ${\mathbb{S}}^{n-1}$, then ${\Sigma }_{L}$ is a contractible $n$–manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced ${\ell }^{2}$–homology except in the middle dimension) in the case of ${\Sigma }_{L}$ where $L$ is a triangulation of ${\mathbb{S}}^{n-1}$. The program succeeds when $n\le 4$. This implies the Charney–Davis Conjecture on flag triangulations of ${\mathbb{S}}^{3}$. It also implies the following special case of the Hopf–Chern Conjecture: every closed 4–manifold with a nonpositively curved, piecewise Euclidean, cubical structure has nonnegative Euler characteristic. Our methods suggest the following generalization of the Singer Conjecture.

Conjecture: If a discrete group $G$ acts properly on a contractible $n$–manifold, then its ${\ell }^{2}$–Betti numbers ${b}_{i}^{\left(2\right)}\left(G\right)$ vanish for $i>n∕2$.

Keywords
Coxeter group, aspherical manifold, nonpositive curvature, $\ell^2$–homology, $\ell^2$–Betti numbers
Mathematical Subject Classification 2000
Primary: 58G12
Secondary: 20F55, 57S30, 20F32, 20J05
Publication
Received: 1 September 2000
Revised: 13 December 2000
Accepted: 31 January 2001
Published: 2 February 2001
Proposed: Walter Neumann
Seconded: Steve Ferry, Ralph Cohen
Authors
 Michael W Davis Department of Mathematics The Ohio State University Columbus Ohio 43210 USA Boris Okun Department of Mathematics Vanderbilt University Nashville Tennessee 37240 USA