The fattened Davis complex and weighted L2–(co)homology of Coxeter groups

Mogilski, Wiktor

doi:10.2140/agt.2016.16.2067

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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

DOI: 10.2140/agt.2016.16.2067

Abstract

This article consists of two parts. First, we propose a program to compute the weighted $L^{2}$ –(co)homology of the Davis complex by considering a thickened version of this complex. The program proves especially successful provided that the weighted $L^{2}$ –(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.

Keywords

weighted L^2 cohomology, fattened Davis complex, Coxeter groups, Singer conjecture

Mathematical Subject Classification 2010

Primary: 20F55

Secondary: 20F65, 53C23, 57M07, 58J22, 46L10

References

Publication

Received: 24 March 2015

Revised: 27 October 2015

Accepted: 12 November 2015

Published: 12 September 2016

Authors

Wiktor J Mogilski
Department of Mathematical Sciences Binghamton University PO Box 6000 Binghamton, NY 13902-6000 USA

Volume 16, issue 4 (2016)