Edge subdivisions and the $L^2$-homology of right-angled Coxeter groups

TL;DR

This paper investigates the $L^2$-homology of Davis complexes associated with right-angled Coxeter groups, establishing conditions under which the homology vanishes outside the middle dimension, and providing counterexamples to related conjectures.

Contribution

It introduces conditions for $L^2$-homology vanishing under edge subdivisions and verifies Singer's conjecture for specific barycentric subdivisions, also constructing counterexamples.

Findings

Verified Singer's conjecture for barycentric subdivisions of simplices

Established conditions preserving $L^2$-homology vanishing under edge subdivision

Constructed explicit counterexamples to a torsion growth analogue

Abstract

If $L$ is a flag triangulation of $S^{n-1}$, then the Davis complex $\Sigma_L$ for the associated right-angled Coxeter group $W_L$ is a contractible $n$-manifold. A special case of a conjecture of Singer predicts that the $L^2$-homology of such $\Sigma_L$ vanishes outside the middle dimension. We give conditions which guarantee this vanishing is preserved under edge subdivision of $L$. In particular, we verify Singer's conjecture when $L$ is the barycentric subdivision of the boundary of an $n$-simplex, and for general barycentric subdivisions of triangulations of $S^{2n-1}$. Using this, we construct explicit counterexamples to a torsion growth analogue of Singer's conjecture.