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The homological torsion of PSL$ _2$ of the imaginary quadratic integers

Author: Alexander D. Rahm
Journal: Trans. Amer. Math. Soc. 365 (2013), 1603-1635
MSC (2010): Primary 11F75, 22E40, 57S30; Secondary 55N91, 19L47, 55R35
Published electronically: August 7, 2012
MathSciNet review: 3003276
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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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Additional Information

Alexander D. Rahm
Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel
Address at time of publication: Department of Mathematics, National University of Ireland at Galway, University Road, Galway, Ireland

Received by editor(s): May 16, 2011
Received by editor(s) in revised form: August 13, 2011
Published electronically: August 7, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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