The weighted Singer conjecture for Coxeter groups in dimensions three and four

TL;DR

This paper proves the weighted Singer conjecture for Coxeter groups in dimensions three and four, establishing new results under specific geometric conditions related to the nerve of the Coxeter group.

Contribution

It extends the weighted Singer conjecture to dimensions three and four for Coxeter groups with particular nerve structures, including flag triangulations of 3-manifolds.

Findings

Proves the weighted Singer conjecture in dimension three for certain Coxeter groups.

Extends the conjecture to dimension four with additional restrictions.

Establishes the conjecture for Coxeter groups with nerve as a flag triangulation of a 3-manifold.

Abstract

Given a Coxeter system $(W,S)$ there is a contractible simplicial complex $\Sigma$ called the Davis complex on which $W$ acts properly and cocompactly. In an article of Dymara, the weighted $L^2$-(co)homology groups of $\Sigma$ were defined, and in an article of Davis-Dymara-Januszkiewicz-Okun, the Singer conjecture for Coxeter groups was appropriately formulated for weighted $L^2$-(co)homology theory. In this article, we prove the weighted version of the Singer conjecture 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 additional restrictions. We then prove a general version of the conjecture where the nerve of the Coxeter group is assumed to be a flag triangulation of a $3$-manifold.