On the three-dimensional Singer Conjecture for Coxeter groups

TL;DR

This paper proves the Singer conjecture for Davis complexes associated with Coxeter groups whose nerve triangulates a 2-sphere, linking it to classical reflection groups acting on hyperbolic 3-space.

Contribution

It establishes the Singer conjecture for a specific class of Coxeter groups by connecting it to Andreev's theorem on reflection groups in hyperbolic space.

Findings

Singer conjecture holds for Coxeter groups with nerve triangulating S^2

Connection established between Davis complex properties and hyperbolic reflection groups

Provides conditions under which the conjecture is valid in this setting

Abstract

We give a proof of the Singer conjecture (on the vanishing of reduced $\ell^2$-homology except in the middle dimension) for the Davis Complex $\Sigma$ associated to a Coxeter system $(W,S)$ whose nerve $L$ is a triangulation of $\mathbb{S}^2$. We show that it follows from a theorem of Andreev, which gives the necessary and sufficient conditions for a classical reflection group to act on $\mathbb{H}^3$.