## The homological torsion of PSL_2 of the imaginary quadratic integers

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2013##### Author

Rahm, Alexander D.

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##### Recommended Citation

Rahm, Alexander D. (2013) 'The homological torsion of PSL_2 of the imaginary quadratic integers'. Trans. Amer. Math. Soc, 365 (3):1603-1635.

##### Abstract

The Bianchi groups are the groups (P)SL2 over a ring of integers in an imaginary
quadratic number field. 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 satisfied by the Bianchi
groups, we obtain the K-theory of their reduced C*-algebras in terms of isomorphic images of
their equivariant K-homology.