ℓ2-homology of right-angled Coxeter groups based on barycentric subdivisions

Associated to any finite flag complex L there is a right-angled Coxeter group WL and a contractible cubical complex ΣL on which WL acts properly and cocompactly, and such that the link of each vertex is L. It follows that if L is a triangulation of Sn−1, then ΣL is a contractible n-manifold. We establish vanishing (in a certain range) of the reduced ℓ2-homology of ΣL in the case where L is the barycentric subdivision of a cellulation of a manifold. In particular, we prove the Singer Conjecture (on the vanishing of the reduced ℓ2-homology except in the middle dimension) in the case of ΣL where L is the barycentric subdivision of a cellulation of Sn−1, n=6,8.