Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 16, Number 4 (2016), 2067-2105.
The fattened Davis complex and weighted $L^2$–(co)homology of Coxeter groups
This article consists of two parts. First, we propose a program to compute the weighted –(co)homology of the Davis complex by considering a thickened version of this complex. The program proves especially successful provided that the weighted –(co)homology of certain infinite special subgroups of the corresponding Coxeter group vanishes in low dimensions. We then use our complex to perform computations for many examples of Coxeter groups. Second, we prove the weighted Singer conjecture for Coxeter groups in dimension three under the assumption that the nerve of the Coxeter group is not dual to a hyperbolic simplex, and in dimension four under the assumption that the nerve is a flag complex. We then prove a general version of the conjecture in dimension four where the nerve of the Coxeter group is assumed to be a flag triangulation of a –manifold.
Algebr. Geom. Topol., Volume 16, Number 4 (2016), 2067-2105.
Received: 24 March 2015
Revised: 27 October 2015
Accepted: 12 November 2015
First available in Project Euclid: 28 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 57M07: Topological methods in group theory 58J22: Exotic index theories [See also 19K56, 46L05, 46L10, 46L80, 46M20] 46L10: General theory of von Neumann algebras
Mogilski, Wiktor. The fattened Davis complex and weighted $L^2$–(co)homology of Coxeter groups. Algebr. Geom. Topol. 16 (2016), no. 4, 2067--2105. doi:10.2140/agt.2016.16.2067. https://projecteuclid.org/euclid.agt/1511895908