Associated to any finite flag complex L there is a right-angled Coxeter group W L and a cubical complex Σ L on which W L acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of Σ L is L and (2) Σ L is contractible. It follows that if L is a triangulation of S n−1 , then Σ L is a contractible n-manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced 2 -homology except in the middle dimension) in the case of Σ L where L is a triangulation of S n−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 2 -Betti numbers b

Given a Coxeter system .W; S / and a positive real multiparameter q, we study the "weighted L 2 -cohomology groups," of a certain simplicial complex † associated to .W; S /. These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to .W; S / and the multiparameter q. They have a "von Neumann dimension" with respect to the associated "Hecke-von Neumann algebra" N q . The dimension of the i -th cohomology group is denoted b For a certain range of q, we calculate these cohomology groups as modules over N q and obtain explicit formulas for the b i q . †/. The range of q for which our calculations are valid depends on the region of convergence of the growth series of W . Within this range, we also prove a Decomposition Theorem for N q , analogous to a theorem of L Solomon on the decomposition of the group algebra of a finite Coxeter group.

The action dimension of a discrete group Γ is the smallest dimension of a contractible manifold which admits a proper action of Γ. Associated to any flag complex L there is a right-angled Artin group, A L . We compute the action dimension of A L for many L. Our calculations come close to confirming the conjecture that if an ℓ 2 -Betti number of A L in degree l is nonzero, then the action dimension of A L is ≥ 2l.

We compute:• the cohomology with group ring coefficients of Artin groups (or actually, of their associated Salvetti complexes), Bestvina-Brady groups, and graph products of groups,• the L 2 -Betti numbers of Bestvina-Brady groups and of graph products of groups,• the weighted L 2 -Betti numbers of graph products of Coxeter groups.In the case of arbitrary graph products there is an additional proviso: either all factors are infinite or all are finite. (However, for graph products of Coxeter groups this proviso is unnecessary.)AMS classification numbers. Primary: 20F36, 20F55, 20F65, 20J06, 55N25 Secondary: 20E42, 57M07.

For any Coxeter group W , we define a filtration of H * (W; ZW) by W -submodules and then compute the associated graded terms. More generally, if U is a CW complex on which W acts as a reflection group we compute the associated graded terms for H * (U) and, in the case where the action is proper and cocompact, for H * c (U).