Vanishing theorems and conjectures for the $\ell^2$–homology of right-angled Coxeter groups

Abstract

Associated to any finite flag complex $L$ there is a right-angled Coxeter group $W_L$ and a cubical complex ${\Sigma }_L$ on which $W_L$ acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of ${\Sigma }_L$ is $L$ and (2) ${\Sigma }_L$ is contractible. It follows that if $L$ is a triangulation of ${\mathbb{S}}^{n-1}$, then ${\Sigma }_L$ is a contractible $n$–manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced ${\ell }^2$–homology except in the middle dimension) in the case of ${\Sigma }_L$ where $L$ is a triangulation of ${\mathbb{S}}^{n-1}$. The program succeeds when $n\le 4$. This implies the Charney–Davis Conjecture on flag triangulations of ${\mathbb{S}}^3$. It also implies the following special case of the Hopf–Chern Conjecture: every closed 4–manifold with a nonpositively curved, piecewise Euclidean, cubical structure has nonnegative Euler characteristic. Our methods suggest the following generalization of the Singer Conjecture.

Conjecture: If a discrete group $G$ acts properly on a contractible $n$–manifold, then its ${\ell }^2$–Betti numbers $b_i^{\left(2\right)}\left(G\right)$ vanish for $i>n∕2$.