On the center-valued Atiyah conjecture for L2-Betti numbers

TL;DR

This paper extends the Atiyah conjecture to center-valued dimensions of L2-Betti numbers, proving it for a broad class of groups and linking it to the structure of division closures in operator rings.

Contribution

It introduces a center-valued version of the Atiyah conjecture, proves it for groups in Linnell's class C, and applies approximation techniques to extend results to residually C groups.

Findings

Proves the conjecture for all groups in Linnell's class C.

Establishes the structure of the division closure D(QG) as a semisimple Artinian ring.

Extends the results to many residually C groups, including certain free and braid groups.

Abstract

The so-called Atiyah conjecture states that the von Neumann dimensions of the L2-homology modules of free G-CW-complexes belong to a certain set of rational numbers, depending on the finite subgroups of G. In this article we extend this conjecture to a statement for the center-valued dimensions. We show that the conjecture is equivalent to a precise description of the tructure as a semisimple Artinian ring of the division closure D(QG) of Q[G] in the ring of affiliated operators. We prove the conjecture for all groups in Linnell's class C, containing in particular free-by-elementary amenable groups. The center-valued Atiyah conjecture states that the center-valued L2-Betti numbers of finite free G-CW-complexes are contained in a certain discrete subset of the center of C[G], the one generated as an additive group by the center-valued traces of all projections in C[H], where H runs through the finite subgroups of G. Finally, we use the approximation theorem of Knebusch for the center-valued $L^2$-Betti numbers to extend the result to many groups which are residually in C, in particular for finite extensions of products of free groups and of pure braid groups.