Weighted $L^2$–cohomology of Coxeter groups based on barycentric subdivisons

Abstract

Associated to any finite flag complex $L$ there is a right-angled Coxeter group $W_L$ and a contractible cubical complex ${\Sigma }_L$ (the Davis complex) on which $W_L$ 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 ${\Sigma }_L$ is a contractible homology manifold. We prove a generalized version of the Singer Conjecture (on the vanishing of the reduced weighted $L_q^2$–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.