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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Finite group extensions and the Atiyah conjecture
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by Peter Linnell and Thomas Schick
J. Amer. Math. Soc. 20 (2007), 1003-1051
Published electronically: March 14, 2007


The Atiyah conjecture for a discrete group $G$ states that the $L^2$-Betti numbers of a finite CW-complex with fundamental group $G$ are integers if $G$ is torsion-free, and in general that they are rational numbers with denominators determined by the finite subgroups of $G$. Here we establish conditions under which the Atiyah conjecture for a torsion-free group $G$ implies the Atiyah conjecture for every finite extension of $G$. The most important requirement is that $H^*(G,\mathbb {Z}/p)$ is isomorphic to the cohomology of the $p$-adic completion of $G$ for every prime number $p$. An additional assumption is necessary e.g. that the quotients of the lower central series or of the derived series are torsion-free. We prove that these conditions are fulfilled for a certain class of groups, which contains in particular Artin’s pure braid groups (and more generally fundamental groups of fiber-type arrangements), free groups, fundamental groups of orientable compact surfaces, certain knot and link groups, certain primitive one-relator groups, and products of these. Therefore every finite, in fact every elementary amenable extension of these groups satisfies the Atiyah conjecture, provided the group does. As a consequence, if such an extension $H$ is torsion-free, then the group ring $\mathbb {C}H$ contains no non-trivial zero divisors, i.e. $H$ fulfills the zero-divisor conjecture. In the course of the proof we prove that if these extensions are torsion-free, then they have plenty of non-trivial torsion-free quotients which are virtually nilpotent. All of this applies in particular to Artin’s full braid group, therefore answering question B6 on Our methods also apply to the Baum-Connes conjecture. This is discussed by Thomas Schick in his preprint “Finite group extensions and the Baum-Connes conjecture”, where for example the Baum-Connes conjecture is proved for the full braid groups.
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Bibliographic Information
  • Peter Linnell
  • Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123
  • MR Author ID: 114455
  • Email:
  • Thomas Schick
  • Affiliation: Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany
  • MR Author ID: 635784
  • Email:
  • Received by editor(s): May 31, 2005
  • Published electronically: March 14, 2007
  • Additional Notes: The first author was partially supported by SFB 478, Münster
    Research of the second author was funded by DAAD (German Academic Exchange Agency)
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 20 (2007), 1003-1051
  • MSC (2000): Primary 55N25, 16S34, 57M25
  • DOI:
  • MathSciNet review: 2328714