Coxeter Groups, Ruins, and Weighted $L^2$-cohomology

TL;DR

This paper investigates the weighted $L^2$-cohomology of Coxeter groups, establishing concentration and vanishing results, and extends previous computations to new classes of Coxeter groups using complex and conjecture resolution techniques.

Contribution

It introduces new vanishing theorems for weighted $L^2$-cohomology of Coxeter groups and extends existing computations to higher-dimensional cases.

Findings

Weighted $L^2$-cohomology is concentrated in low dimensions for certain parameter ranges.

New vanishing results are proved for specific low-dimensional Coxeter groups.

Extensions of Davis and Okun's computations to higher-dimensional cases are achieved.

Abstract

Given a Coxeter system $(W,S)$ and a multiparameter $\mathbf{q}$ of real numbers indexed by $S$, one can define the weighted $L^2$-cohomology groups and associate to them a nonnegative real number called the weighted $L^2$-Betti number. We show that for ranges of $\mathbf{q}$ depending on certain subgroups of $W$, the weighted $L^2$-cohomology groups of $W$ are concentrated in low dimensions. We then prove new vanishing results for the weighted $L^2$-cohomology of certain low-dimensional Coxeter groups. Our arguments rely on computing the $L^2$-cohomology of certain complexes called ruins, as well as the resolution of the Strong Atiyah Conjecture for hyperbolic Coxeter groups. We conclude by extending to the weighted setting the computations of Davis and Okun for the case where the nerve of a right-angled Coxeter group is the barycentric subdivision of a PL-cellulation of an $(n-1)$-manifold with $n=6,8$.