On the three-dimensional Singer Conjecture for Coxeter groups

We give a proof of the Singer conjecture (on the vanishing of reduced ${\mathrm{\ell }}^2$-homology except in the middle dimension) for the Davis Complex $\mathrm{\Sigma }$ 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 ${\mathbb{H}}^3$.