14.00 – 14.45 : François Thilmany (KU Leuven) « Finding ping-pong partners for finite subgroups of linear groups »

In his paper on free subgroups of linear groups, Tits proved his famous alternative: a linear group is either virtually solvable, or contains a free subgroup. Since then, Tits’ work has been generalized and applied in many different ways. 
One remaining open question in this subject is the one asked by de la Harpe and his collaborators: let $G$ be a semisimple Lie group without compact factors and with trivial center, and let $\Gamma$ be a Zariski-dense subgroup of $G$. Given a prescribed finite subset $F$ of $G$, is it always possible to find an element $\gamma \in \Gamma$ such that for any $h \in F$, the subgroup generated by $h$ and $\gamma$ is freely generated? (If so, we say $h$ and $\gamma$ are ping-pong partners.)  
In this talk, we will discuss a variant of the question of de la Harpe, where $F$ is a finite set of finite subgroups $H_i$ of $G$. Using careful refinements of the main steps of Tits’ proof of the alternative (which we will recall), we give sufficient conditions for the existence of ping-pong partners for the $H_i$ in any Zariski-dense subgroup $\Gamma$. 
We will then show that these conditions are satisfied for products of copies of $\mathrm{SL}_n$ over division $\mathbb{R}$-algebras. 
The existence of such free products has applications in the theory of integral group rings of finite groups, which will be briefly mentioned.
Joint work with G. Janssens and D. Temmerman. 

15.00 – 15.45 : Milan Donvil (Ecole normale supérieure) « The quantum groups behind W*-superrigidity » 

To any countable group, one can associate its group von Neumann algebra, which is the closure of the group ring in a weak topology. A group is called W*-superrigid if its group von Neumann algebra cannot be isomorphic to the von Neumann algebra of another nonisomorphic group. One says that the group is ‘completely recoverable’ from its von Neumann algebra. To prove W*-superrigidity, one actually needs tools from the theory of compact quantum groups, which are von Neumann algebras with additional structure. Since group von Neumann algebras are in particular compact quantum groups, it is natural to ask if there are groups which are also superrigid within this larger class. I will explain the link between W*-superrigidity and compact quantum groups, as well as present a resent work of Stefaan Vaes and me which provides the first ‘quantum W*-superrigid’ (quantum) groups. 

16.15 – 17.00 : Pegah Pournajafi (Collège de France) « Quantum automorphism groups of 0-hyperbolic graphs »Quantum groups and graph theory may seem like distant areas, yet intriguing connections emerge when they intersect. After an introduction to the notion of quantum automorphism groups of finite graphs, we will focus on 0-hyperbolic graphs and a computation of their quantum automorphism group. If time permits, we will also show how their quantum symmetries can be fully understood through their classical properties, due to their structural constraints. This talk is based on joint work with Amaury Freslon and Paul Meunier.