  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.
