Vanishing theorems and conjectures for the ℓ2–homology of right-angled Coxeter groups

Associated to any finite flag complex $L$ there is a right-angled Coxeter group $W_{L}$ and a cubical complex $Σ_{L}$ on which $W_{L}$ acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of $Σ_{L}$ is $L$ and (2) $Σ_{L}$ is contractible. It follows that if $L$ is a triangulation of $S^{n - 1}$ , then $Σ_{L}$ 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 $Σ_{L}$ where $L$ is a triangulation of $S^{n - 1}$ . The program succeeds when $n \leq 4$ . 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.

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, $\ell^2$–homology, $\ell^2$–Betti numbers

Mathematical Subject Classification 2000

Primary: 58G12

Secondary: 20F55, 57S30, 20F32, 20J05

References

Forward citations

Publication

Received: 1 September 2000

Revised: 13 December 2000

Accepted: 31 January 2001

Published: 2 February 2001

Proposed: Walter Neumann

Seconded: Steve Ferry, Ralph Cohen

Authors

Michael W Davis
Department of Mathematics The Ohio State University Columbus Ohio 43210 USA

Boris Okun
Department of Mathematics Vanderbilt University Nashville Tennessee 37240 USA

Volume 5, issue 1 (2001)

Michael W Davis and Boris Okun