Integrality of L2-Betti numbers

Abstract

The Atiyah conjecture predicts that the L2-Betti numbers of a finite CW-complex with torsion-free fundamental group are integers. We show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of directed systems of groups for which it is true. As a corollary it holds for residually torsion-free solvable groups, e.g. for pure braid groups or for positive 1-relator groups with torsion free abelianization. Putting everything together we establish a new 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 elementary amenable quotient, and under direct and inverse limits of directed systems. This is a corrected version of an older paper with the same title. The proof of Proposition 2.1 of the earlier version contains a gap, as was pointed out to me by Pere Ara. This gap could not be fixed. Consequently, in this new version everything based on this result had to be removed. Please take the errata to "L2-determinant class and approximation of L2-Betti numbers" into account, which are added, rectifying some unproved statements about "amenable extension". As a consequence, throughout, amenable extensions should be extensions with normal subgroups.