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Denote by Q(sqrt{-m}), with m a square-free positive integer, an imaginaryquadratic number field, and by A its ring of integers. The Bianchi groups arethe groups SL_2(A). We reveal a correspondence between the homological torsionof the Bianchi groups and new geometric invariants, which are effectivelycomputable thanks to their action on hyperbolic space. We expose a noveltechnique, the torsion subcomplex reduction, to obtain these invariants. We useit to explicitly compute the integral group homology of the Bianchi groups.Furthermore, this correspondence facilitates the computation of the equivariantK-homology of the Bianchi groups. By the Baum/Connes conjecture, which isverified by the Bianchi groups, we obtain the K-theory of their reducedC*-algebras in terms of isomorphic images of their equivariant K-homology.

The homological torsion of PSL_2 of the imaginary quadratic integers