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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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


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
  • Email:,
  • Received by editor(s): May 16, 2011
  • Received by editor(s) in revised form: August 13, 2011
  • Published electronically: August 7, 2012
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 1603-1635
  • MSC (2010): Primary 11F75, 22E40, 57S30; Secondary 55N91, 19L47, 55R35
  • DOI:
  • MathSciNet review: 3003276