  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).}
