Geometry & Topology

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

Boris Okun

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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

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

Received: 15 March 2004
Revised: 3 August 2004
Accepted: 11 July 2004
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58G12
Secondary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15] 57S30: Discontinuous groups of transformations 20F32 20J05: Homological methods in group theory

Coxeter group aspherical manifold barycentric subdivision weighted $L^2$–cohomology Tomei manifold Singer conjecture


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.

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  • M W Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math. 117 (1983) 293–324
  • M W Davis, Coxeter groups and aspherical manifolds, from: “Algebraic topology (Aarhus 1982)”, Lecture Notes in Math. 1051, Springer, New York (1984) 197–221
  • M W Davis, Some aspherical manifolds, Duke Math. J. 55 (1987) 105–139
  • M W Davis, Nonpositive curvature and reflection groups, from: “Handbook of Geometric Topology”, (R Daverman, R Sher, editors), Elsevier, Amsterdam (2002) 373–422
  • M W Davis, T Januszkiewicz, J Dymara, B L Okun, Weighted $L^2$–cohomology of Coxeter groups.
  • M W Davis, B L Okun, $\ell_2$–homology of Coxeter groups based on barycentric subdivisions, Topol. Appl. 140 (2004) 197–202
  • J Dymara, Thin buildings, preprint
  • C Tomei, The topology of isospectral manifolds of tridiagonal matrices, Duke Math. J. 51 (1984) 981–996