Geometry & Topology, Vol. 5 (2001)
Paper no. 2, pages 7-74.
Vanishing theorems and conjectures for the L^2-homology of right-angled Coxeter groups
Michael W Davis, Boris Okun
Abstract.
Associated to any finite flag complex L there is a right-angled
Coxeter group W_L and a cubical complex \Sigma _L on which W_L acts
properly and cocompactly. Its two most salient features are that (1)
the link of each vertex of \Sigma _L is L and (2) \Sigma _L is
contractible. It follows that if L is a triangulation of S^{n-1}, then
\Sigma _L is a contractible n-manifold. We describe a program for
proving the Singer Conjecture (on the vanishing of the reduced
L^2-homology except in the middle dimension) in the case of \Sigma _L
where L is a triangulation of S^{n-1}. The program succeeds when n <
5. This implies the Charney-Davis Conjecture on flag triangulations of
S^3. It also implies the following special case of the Hopf-Chern
Conjecture: every closed 4-manifold with a nonpositively curved,
piecewise Euclidean, cubical structure has nonnegative Euler
characteristic. Our methods suggest the following generalization of
the Singer Conjecture. \par {Conjecture:} If a discrete group G acts
properly on a contractible n-manifold, then its L^2-Betti numbers
b_i^{(2)} (G)$ vanish for i>n/2.
Keywords.
Coxeter group, aspherical manifold, nonpositive curvature,
L^2-homology, L^2-Betti numbers
AMS subject classification.
Primary: 58G12.
Secondary: 20F55, 57S30, 20F32, 20J05.
DOI: 10.2140/gt.2001.5.7
E-print: arXiv:math.GR/0102104
Submitted to GT on 1 September 2000.
(Revised 13 December 2000.)
Paper accepted 31 January 2001.
Paper published 02 February 2001.
Notes on file formats
Michael W Davis, Boris Okun
Department of Mathematics, The Ohio State University
Columbus, OH 43210, USA
Department of Mathematics, Vanderbilt University
Nashville, TN 37240, USA
Email: mdavis@math.ohio-state.edu, okun@math.vanderbilt.edu
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