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 WL and a contractible cubical complex ΣL (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 ΣL is a contractible homology manifold. We prove a generalized version of the Singer Conjecture (on the vanishing of the reduced weighted Lq2–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
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Publication
Received: 15 March 2004
Revised: 3 August 2004
Accepted: 11 July 2004
Published: 7 August 2004
Proposed: Benson Farb
Seconded: Walter Neumann, Steven Ferry
Authors
Boris Okun
Department of Mathematical Sciences
University of Wisconsin–Milwaukee
Milwaukee
Wisconsin 53201
USA