Weighted L²-cohomology of Coxeter groups based on barycentric subdivisons

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Associated to any finite flag complex L there is a right-angled Coxeter group Wₗ and a contractible cubical complex Sigmaₗ (the Davis complex) on which Wₗ acts properly and cocompactly, and such that the link of each vertex is L. It follows that if L is a generalized homology sphere, then Sigmaₗ is a contractible homology manifold. We prove a generalized version of the Singer Conjecture (on the vanishing of the reduced weighted L²q-cohomology above the middle dimension) for the right-angled Coxeter groups based on barycentric subdivisions in even dimensions. We also prove this conjecture for the groups based on the barycentric subdivision of the boundary complex of a simplex.