Geometry & Topology
- Geom. Topol.
- Volume 5, Number 1 (2001), 7-74.
Vanishing theorems and conjectures for the $\ell^2$–homology of right-angled Coxeter groups
Associated to any finite flag complex there is a right-angled Coxeter group and a cubical complex on which acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of is and (2) is contractible. It follows that if is a triangulation of , then is a contractible –manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced –homology except in the middle dimension) in the case of where is a triangulation of . The program succeeds when . This implies the Charney–Davis Conjecture on flag triangulations of . 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.
Conjecture: If a discrete group acts properly on a contractible –manifold, then its –Betti numbers vanish for .
Geom. Topol., Volume 5, Number 1 (2001), 7-74.
Received: 1 September 2000
Revised: 13 December 2000
Accepted: 31 January 2001
First available in Project Euclid: 21 December 2017
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Davis, Michael W; Okun, Boris. Vanishing theorems and conjectures for the $\ell^2$–homology of right-angled Coxeter groups. Geom. Topol. 5 (2001), no. 1, 7--74. doi:10.2140/gt.2001.5.7. https://projecteuclid.org/euclid.gt/1513882983