The homological torsion of PSL₂ of the imaginary quadratic integers

Rahm, Alexander

doi:10.1090/S0002-9947-2012-05690-X

The homological torsion of PSL$_2$ of the imaginary quadratic integers
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Trans. Amer. Math. Soc. 365 (2013), 1603-1635 Request permission

Abstract:

The Bianchi groups are the groups (P)$\mathrm {SL_2}$ 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.

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