Volume 8, issue 3 (2004)

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

Boris Okun

Geometry & Topology 8 (2004) 1032–1042
 arXiv: math.GR/0408149
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.

Keywords
Coxeter group, aspherical manifold, barycentric subdivision, weighted $L^2$–cohomology, Tomei manifold, Singer conjecture
Mathematical Subject Classification 2000
Primary: 58G12
Secondary: 20F55, 57S30, 20F32, 20J05