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


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