Integrality of L2L^2-Betti numbers |
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Authors: | Thomas Schick |
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Institution: | (1) FB Mathematik, Universit?t Münster, Einsteinstr. 62, 48149 Münster, Germany (e-mail: thomas.schick@math.uni-muenster.de) , DE |
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Abstract: | The Atiyah conjecture predicts that the -Betti numbers of a finite CW-complex with torsion-free fundamental group are integers. We establish the Atiyah conjecture, under the condition that it
holds for G and that is a normal subgroup, for amalgamated free products . Here F is a free group and is an arbitrary semi-direct product. This includes free products G*F and semi-direct products . We also show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of
groups for which it is true. As a corollary it holds for positive 1-relator groups with torsion free abelianization. Putting
everything together we establish a new (bigger) class of groups for which the Atiyah conjecture holds, which contains all
free groups and in particular is closed under taking subgroups, direct sums, free products, extensions with torsion-free elementary
amenable quotient or with free quotient, and under certain direct and inverse limits.
Received: 22 August 1998/ Revised: 10 Jannary 2000 / Published online: 28 June 2000 |
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Keywords: | Mathematics Subject Classification (1991): 55N25 16S34 46L50 |
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