## Geometry & Topology

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

Boris Okun

#### 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.

#### Article information

Source
Geom. Topol., Volume 8, Number 3 (2004), 1032-1042.

Dates
Revised: 3 August 2004
Accepted: 11 July 2004
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883461

Digital Object Identifier
doi:10.2140/gt.2004.8.1032

Mathematical Reviews number (MathSciNet)
MR2087077

Zentralblatt MATH identifier
1062.58026

#### Citation

Okun, Boris. Weighted $L^2$–cohomology of Coxeter groups based on barycentric subdivisons. Geom. Topol. 8 (2004), no. 3, 1032--1042. doi:10.2140/gt.2004.8.1032. https://projecteuclid.org/euclid.gt/1513883461

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