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 are rational with denominators determined by the finite subgroups of G in general. Here we establish conditions under which the Atiyah conjecture for a group G implies the Atiyah conjecture for every finite extension of G. The most important requirement is that the cohomology $H^*(G,\mathbb{Z}/p)$ 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 class of groups which contains Artin's pure braid groups, free groups, surfaces groups, certain link groups and one-relator groups. Therefore every finite, in fact every elementary amenable extension of these groups satisfies the Atiyah 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 http://www.grouptheory.info . Our methods also apply to the Baum-Connes conjecture. This is discussed in arXiv:math/0209165 "Finite group extensions and the Baum-Connes conjecture", where the Baum-Connes conjecture is proved e.g. for the full braid group.