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Finite group extensions and the Atiyah conjecture

Authors: Peter Linnell and Thomas Schick
Journal: J. Amer. Math. Soc. 20 (2007), 1003-1051
MSC (2000): Primary 55N25, 16S34, 57M25
Published electronically: March 14, 2007
MathSciNet review: 2328714
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Abstract: 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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Additional Information

Peter Linnell
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123
MR Author ID: 114455

Thomas Schick
Affiliation: Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany
MR Author ID: 635784

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)
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.