  Denote by Q(sqrt{-m}), with m a square-free positive integer, an imaginary
quadratic number field, and by A its ring of integers. The Bianchi groups are
the groups SL_2(A). 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 expose a novel
technique, the torsion subcomplex reduction, to obtain these invariants. We use
it to explicitly compute the integral group homology of the Bianchi groups.
Furthermore, this correspondence facilitates the computation of the equivariant
K-homology of the Bianchi groups. By the Baum/Connes conjecture, which is
verified by the Bianchi groups, we obtain the K-theory of their reduced
C*-algebras in terms of isomorphic images of their equivariant K-homology.
