Thomas Schick: On the center valued Atiyah conjecture for L2-Betti numbers скачать в хорошем качестве

Thomas Schick: On the center valued Atiyah conjecture for L2-Betti numbers

The lecture was held within the framework of the (Junior) Hausdorff Trimester Program Topology and the Workshop: New directions in L2-invariants (04.10.2016) 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 structure 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 L2-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. This is joint work with Anselm Knebusch and Peter Linnel.