Weighted L2–cohomology of Coxeter groups based on barycentric subdivisons

Okun, Boris

doi:10.2140/gt.2004.8.1032

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

Boris Okun

Geometry & Topology 8 (2004) 1032–1042

DOI: 10.2140/gt.2004.8.1032

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 $Σ_{L}$ (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 $Σ_{L}$ 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

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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

Volume 8, issue 3 (2004)