Linnell, P. and Schick, T.: Finite group extensions and the Atiyah conjecture

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 the cohomology $H^*(G,\integers/p)$ 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 $\complexs H$ 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 in Finite group extensions and the Baum-Connes conjecture (also on the arXiv), where the Baum-Connes conjecture is proved e.g. for the full braid group.

MSC: 55N25 (homology with local coefficients), 16S34 (group rings, Laurent rings), 57M25 (knots and links)

Thomas Schick