Homology and K-theory of the Bianchi groups

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 $C^{\ast }$-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).}

References in corpus (1)

• The integral homology of $PS{L}_2$ of imaginary quadratic integers with non-trivial class group

Cited by in corpus (5)

• On Level One Cuspidal Bianchi Modular Forms
• Higher torsion in the Abelianization of the full Bianchi groups
• Genuine Bianchi modular forms of higher level, at varying weight and discriminant
• On the equivariant $K$- and $KO$-homology of some special linear groups
• A refined Bloch group and the third homology of SL_2 of a field