Vanishing theorems and conjectures for the l2-homology of right-angled Coxeter groups

Associated to any finite flag complex L there is a right-angled Coxeter group WL 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 Sn-1, then Sigma(L) is a contractible n-manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced l(2)-homology except in the middle dimension) in the case of Sigma(L) where L is a triangulation of Sn-1. The program succeeds when n <= 4. This implies the Charney-Davis Conjecture on flag triangulations of 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 l(2)-Betti numbers b(i)((2))(C) vanish for i > n/2.