Integrality of L2-Betti numbers

TL;DR

This paper proves the Atiyah conjecture for a broad class of groups, including free groups and residually torsion-free solvable groups, by analyzing limits of groups where the conjecture holds, thus expanding its verified cases.

Contribution

It establishes the Atiyah conjecture for new classes of groups using limits of groups where the conjecture is known to hold, correcting previous gaps and extending the scope of the conjecture.

Findings

The conjecture holds for residually torsion-free solvable groups.

The class of groups satisfying the conjecture is closed under various group operations.

The paper corrects previous proofs and clarifies conditions for amenable extensions.

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.