Equivariant $K$-homology for hyperbolic reflection groups

TL;DR

This paper computes the equivariant K-homology of 3D hyperbolic reflection groups, confirming torsion-freeness of their K-theory groups and introducing a new algebraic criterion for such analyses.

Contribution

It provides the first integral computations of K-theory for these groups and introduces a novel algebraic method for checking torsion-freeness.

Findings

K-theory groups are torsion-free for all considered groups

Integral computations extend previous rational results

New algebraic criterion for torsion-freeness

Abstract

We compute the equivariant $K$-homology of the classifying space for proper actions, for compact 3-dimensional hyperbolic reflection groups. This coincides with the topological $K$-theory of the reduced $C^\ast$-algebra associated to the group, via the Baum-Connes conjecture. We show that, for any such reflection group, the associated $K$-theory groups are torsion-free. As a result we can promote previous rational computations to integral compu- tations. Our proof relies on a new efficient algebraic criterion for checking torsion-freeness of K-theory groups, which could be applied to many other classes of groups.