Integrality of $L^2$-Betti numbers

Abstract.

The Atiyah conjecture predicts that the $L^2$-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 $H\lhd G$ is a normal subgroup, for amalgamated free products $G*_{H}(H\rtimes F)$. Here F is a free group and $H\rtimes F$ is an arbitrary semi-direct product. This includes free products G*F and semi-direct products $G\rtimes F$. 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.