# L^{2}-Seminar

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45, Bonn (unless stated otherwise)

Organizer: Holger Kammeyer

## Monday, October 10

15:00 - 16:00 Lukasz Grabwoski: Group ring element with large spectral measure

Abstract: I will describe how to construct, for a given d>1, a group and a group ring element T with the property that for all sufficiently small ε, the spectral measure associated to T of the interval (0,ε) is more than 1/|log(ε)|^{d}. This almost matches the best known general upper bounds on the spectral measure due to Lück and Schick. I will also discuss some potential ideas on how to improve those examples to show that the best known general upper bounds are in fact optimal.

## Friday, October 14

9:30 - 10:30 Jean Raimbault: Prelude to an introduction to the trace formula: the case of SL_{2}(ℤ)

## Monday, October 17

15:00 - 16:00 Werner Müller: Introduction to the trace formula (part 1)

## Tuesday, October 18

10:00 - 11:00 Werner Müller: Introduction to the trace formula (part 2)

## Monday, October 31

15:00 - 16:00 Florian Funke: The Determinant Comparison Problem

Abstract: We discuss an open, purely algebraic statement about polytopes of matrices over group rings and point out its relevance in the context of Friedl-Lück’s universal L2-torsion and seminorms on the first cohomology of groups. Alongside we’ll digress to study the integral polytope group itself.

## Monday, November 28

15:00 - 16:00 Vadim Alekseev: Sofic boundaries of groups and coarse geometry of sofic approximations

Abstract: Sofic groups generalise both residually finite and amenable groups, and the concept is central to many important results and conjectures in measured group theory. In the recent work with Martin Finn-Sell, we introduce a topological notion of a sofic boundary attached to a given sofic approximation of a finitely generated group and use it to prove that coarse properties of the approximation (property A, asymptotic coarse embeddability into Hilbert space, geometric property (T)) imply corresponding analytic properties of the group (amenability, a-T-menability

and property (T)), thus generalising ideas and results present in the literature for residually finite groups and their box spaces. Moreover, we generalise coarse rigidity results for box spaces due to Kajal Das, proving that coarsely equivalent sofic approximations of two groups give rise to a uniform measure equivalence between those groups.

## Wednesday, November 30

11:00 - 12:00 Roman Sauer: L^{2}-Approximation for lattices in totally disconnected groups

Abstract: We discuss an approximation theorem for L^{2}-Betti numbers of a sequence of lattices in a totally disconnected group that converges in the sense of invariant random subgroups. Joint work with A. Thom and H. Petersen.

## Monday, December 12

15:00 - 16:00 Wolfgang Lück: Survey on approximating L^{2}-invariants

Abstract: In this talk we discuss some prominent open problems concerning L^{2}-invariants focusing on approximation by towers of finite coverings. We discuss the status of various open conjectures and give some explanations about possible strategies to prove them.