On the three-dimensional Singer Conjecture for Coxeter groups

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 $\mathbb{H}^3$.