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The fattened Davis complex and weighted $L^2$–(co)homology of Coxeter groups

Wiktor J Mogilski

Algebraic & Geometric Topology 16 (2016) 2067–2105

This article consists of two parts. First, we propose a program to compute the weighted L2–(co)homology of the Davis complex by considering a thickened version of this complex. The program proves especially successful provided that the weighted L2–(co)homology of certain infinite special subgroups of the corresponding Coxeter group vanishes in low dimensions. We then use our complex to perform computations for many examples of Coxeter groups. Second, we prove the weighted Singer conjecture for Coxeter groups in dimension three under the assumption that the nerve of the Coxeter group is not dual to a hyperbolic simplex, and in dimension four under the assumption that the nerve is a flag complex. We then prove a general version of the conjecture in dimension four where the nerve of the Coxeter group is assumed to be a flag triangulation of a 3–manifold.

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weighted L^2 cohomology, fattened Davis complex, Coxeter groups, Singer conjecture
Mathematical Subject Classification 2010
Primary: 20F55
Secondary: 20F65, 53C23, 57M07, 58J22, 46L10
Received: 24 March 2015
Revised: 27 October 2015
Accepted: 12 November 2015
Published: 12 September 2016
Wiktor J Mogilski
Department of Mathematical Sciences
Binghamton University
PO Box 6000
Binghamton, NY 13902-6000