Finite group extensions and the Atiyah conjecture

By P. Linnell, Thomas Schick. 2004

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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,Z/p) is isomorphic to the cohomology of the p-adic completion of G for every prime 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, primitive positive one-relator groups, and products of these. Therefore every finite, in fact every elementary amenable extension of these groups satisfies the Atiyah conjecture. As a consequence, if such an extension H is torsion-free then the group ring CH 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 torsionfree, 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 www.grouptheory.org. Our methods also apply to the Baum-Connes conjecture. This is discussed in [47], where for example the Baum-Connes conjecture is proved for the full braid groups. MSC: 55N25 (homology with local coecients), 16S34 (group rings, Laurent rings), 57M25 (knots and links)

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