Countable groups: an operator algebra point of view

20.01.2016, 14:00  –  Campus Golm, Haus 9, Raum 2.22
Institutskolloquium

Georges Skandalis, Sara Azzali

14:00 Georges Skandalis (Paris): "Traces and determinants on non commutative algebras and countable groups"
15:30 Sara Azzali (Potsdam): "Countable groups, covering spaces and spectral invariants"

"Traces and determinants on non commutative algebras and countable groups"
Gromov introduced a class of (countable) groups called sofic groups that are well approachable by permutation groups, in a sense that can be made precise. All profinite groups, which can roughly be viewed as formal limits of finite groups, and all amenable groups, those carrying a kind of invariant averaging operation, are sofic. For the moment, we do not know any non sofic group. A big part of the talk will be dedicated to explaining and illustrating the very natural notion of sofic groups, after which we will turn to traces and determinants. Loosely speaking, Lück’s determinant conjecture predicts that for x in the group ring <tex> \mathbb{Z} G </tex> of a group G, the continuous product of the eigenvalues of <tex>x^*x</tex> is no smaller than 1. Elek and Szabo established Lück’s determinant conjecture for (Gromov’s) sofic groups. We will present a work in collaboration with G. Balci in which we give a new definition of sofic groups and a new proof of this conjecture in terms of traces on the free group. If time allows we will briefly discuss the relation with Atiyah’s problem on the integrality of L 2 -Betti numbers. 

"Countable groups, covering spaces and spectral invariants"
The study of the spectrum of elliptic operators reveals many fascinating relations between analysis and topology. However, when one leaves the realm of compact spaces, the usual analytic or topological invariants become infinite dimensional. We shall focus on non-compact spaces which carry the action of a group, typically covering spaces. A toy model is Rn with the action of the discrete group Zn and quotient space corresponding to the n-torus. To handle the infinite dimensionality coming from the non-compactness, Atiyah introduced a (real-valued) dimension which takes into account the action of the group. In this way he defined the so called L2 -invariants: for example, Betti numbers generalise to L 2 -Betti numbers. In this talk, we will introduce these objects focusing on spectral invariants of geometric operators. More specifically, we will discuss some secondary invariants introduced by Cheeger and Gromov as the L2 -analog of Atiyah-Patodi-Singer’s spectral asymmetry. More specifically, we will discuss an invariant introduced by Atiyah, Patodi and Singer which reflects the asymmetry of the spectrum as well as its L2 -analog introduced by Cheeger and Gromov. We will present some classical applications and a related result obtained in collaboration with Charlotte Wahl.