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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 WL and a cubical complex ΣL on which WL acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of ΣL is L and (2) ΣL is contractible. It follows that if L is a triangulation of Sn1, then ΣL is a contractible n–manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced 2–homology except in the middle dimension) in the case of ΣL where L is a triangulation of Sn1. The program succeeds when n 4. This implies the Charney–Davis Conjecture on flag triangulations of S3. 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 2–Betti numbers bi(2)(G) vanish for i > n2.

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
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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