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

TL;DR

This paper introduces a method to compute weighted $L^2$-(co)homology of Coxeter groups using a thickened Davis complex, enabling explicit calculations of weighted $L^2$-Betti numbers in many cases.

Contribution

It proposes a new approach involving a thickened Davis complex to compute weighted $L^2$-(co)homology, especially when certain subgroups' homology vanishes in low dimensions.

Findings

Successfully computed weighted $L^2$-Betti numbers for various Coxeter groups.

Provided explicit formulas for these Betti numbers in most cases.

Demonstrated the effectiveness of the thickened complex approach in this context.

Abstract

Associated to a Coxeter system $(W,S)$ there is a contractible simplicial complex $\Sigma$ called the Davis complex on which $W$ acts properly and cocompactly by reflections. Given a positive real multiparameter $\mathbf{q}$, one can define the weighted $L^2$-(co)homology groups of $\Sigma$ and associate to them a nonnegative real number called the weighted $L^2$-Betti number. Not much is known about the behavior of these groups when $\mathbf{q}$ lies outside a certain restricted range, and weighted $L^2$-Betti numbers have proven difficult to compute. In this article we propose a program to compute the weighted $L^2$-(co)homology of $\Sigma$ 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 $W$ vanishes in low dimensions. We then use our complex to perform computations for many examples of Coxeter groups, in most cases providing explicit formulas for the weighted $L^2$-Betti numbers.