Integrality of $L^2$-Betti numbers

By Thomas Schick. 2000

54 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/65225a6721fdfb121ecd7aa8277a649392544100

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Finite group extensions and the Atiyah conjecture (2004, result): “In this section, we recall a few more known results about the Atiyah conjecture, proved in particular in [ 47 ] and [11].”
Hyperlinearity, essentially free actions and L2-invariants. The sofic property (2004, background): “The subgroup of S(τ) in Definition 2.2 is indeed a normal subgroup, so S(τ/ω) is well-defined.”