Homology and K-theory of the Bianchi groups

TL;DR

This paper establishes a link between the homological torsion of Bianchi groups and geometric invariants, enabling explicit computation of their homology and $K$-theory, with applications to orbifold cohomology.

Contribution

It introduces a new method to compute the homology and $K$-theory of Bianchi groups using their action on hyperbolic space, confirming the Baum/Connes conjecture for these groups.

Findings

Explicit computation of integral group homology of Bianchi groups

Determination of their equivariant $K$-homology and $K$-theory of reduced $C^*$-algebras

Application to Chen/Ruan orbifold cohomology

Abstract

We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We use it to explicitly compute their integral group homology and equivariant $K$-homology. By the Baum/Connes conjecture, which holds for the Bianchi groups, we obtain the $K$-theory of their reduced $C^*$-algebras in terms of isomorphic images of the computed $K$-homology. We further find an application to Chen/Ruan orbifold cohomology. % {\it To cite this article: Alexander D. Rahm, C. R. Acad. Sci. Paris, Ser. I +++ (2011).}