Geometry & Topology

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

Michael W Davis and Boris Okun

Full-text: Open access

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 Sn1, then ΣL is a contractible n–manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced 2–homology except in the middle dimension) in the case of ΣL where L is a triangulation of Sn1. The program succeeds when n4. 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 2–Betti numbers bi(2)(G) vanish for i>n2.

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

Subjects
Primary: 58G12
Secondary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15] 57S30: Discontinuous groups of transformations 20F32 20J05: Homological methods in group theory

Keywords
Coxeter group aspherical manifold nonpositive curvature $\ell^2$–homology $\ell^2$–Betti numbers

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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