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

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.