The $\ell^2$-homology of even Coxeter groups

TL;DR

This paper proves that for certain Coxeter groups with nerve as a flag triangulation of the 3-sphere, the reduced -homology of the associated complex vanishes outside the middle dimension, advancing understanding of their topological properties.

Contribution

It establishes the vanishing of reduced -homology for a class of Coxeter groups with nerves triangulating the 3-sphere, linking geometric and algebraic properties.

Findings

-homology vanishes outside the middle dimension

Results apply to Coxeter groups with nerve as a flag triangulation of the 3-sphere

Provides new insights into the topology of Coxeter group complexes

Abstract

Given a Coxeter system (W,S), there is an associated CW-complex, Sigma, on which W acts properly and cocompactly. We prove that when the nerve L of (W,S) is a flag triangulation of the 3-sphere, then the reduced $\ell^2$-homology of Sigma vanishes in all but the middle dimension.