Geometry & Topology

Vanishing theorems and conjectures for the $\ell^2$–homology of right-angled Coxeter groups

Abstract

Associated to any finite flag complex $L$ there is a right-angled Coxeter group $WL$ and a cubical complex $ΣL$ on which $WL$ acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of $ΣL$ is $L$ and (2) $ΣL$ is contractible. It follows that if $L$ is a triangulation of $Sn−1$, then $ΣL$ is a contractible $n$–manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced $ℓ2$–homology except in the middle dimension) in the case of $ΣL$ where $L$ is a triangulation of $Sn−1$. The program succeeds when $n≤4$. This implies the Charney–Davis Conjecture on flag triangulations of $S3$. 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 $ℓ2$–Betti numbers $bi(2)(G)$ vanish for $i>n∕2$.

Article information

Source
Geom. Topol., Volume 5, Number 1 (2001), 7-74.

Dates
Received: 1 September 2000
Revised: 13 December 2000
Accepted: 31 January 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513882983

Digital Object Identifier
doi:10.2140/gt.2001.5.7

Mathematical Reviews number (MathSciNet)
MR1812434

Zentralblatt MATH identifier
1118.58300

Citation

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

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