Séminaire "Groupes et actions"

Les lundis de 14h à 15h à l'Insitut de Mathématiques d'Orsay, salle 2L8. Organisé par Adrien Abgrall et Camille Horbez.

Programme automne 2025

15 septembre: Sasha Bontemps (ENS Lyon), "Subgroup mixing in Baumslag-Solitar groups"

Résumé : Endowed with the Chabauty topology, the space of subgroups of any infinite countable group G is a closed subspace of the Cantor set, equipped with an action by homeomorphisms given by the G-conjugation. We are interested in the dynamics induced by this action on closed G-invariant subspaces. The largest closed subspace without isolated point is an example of such subspace called the perfect kernel of G. In an acylindrically hyperbolic context, Hull, Mynasyan and Osin demonstrated strong mixing properties (namely µ-mixing for a suitable measure µ on G, a strengthening of high topological transitivity). We uncover a radically different situation in the case of non metabelian Baumslag-Solitar groups. For the decomposition of the perfect kernel introduced by Carderi, Gaboriau, Le Maître and Stalder, who proved high topological transitivity on each piece, we show that the conjugation is even µ-mixing in the case of unimodular Baumslag-Solitar groups. On the contrary, when the group is non unimodular, there exists a continuum of measures µ for which the action is µ-mixing only on a single piece of the partition.

22 septembre: Pénélope Azuelos (Bristol), " Group structures on real trees and their products "

Résumé : Median spaces form a broad and increasingly important class of metric spaces, encompassing both CAT(0) cube complexes and real trees. The study of finitely generated groups admitting free transitive (or proper cocompact) actions on discrete median spaces—equivalently, on the 0-skeletons of CAT(0) cube complexes—are an active area of study. In contrast, much less is understood about their continuous analogue: groups acting freely and transitively on connected median spaces. I will present several methods for constructing such actions, focusing on actions on real trees and their products, and discuss some of the surprising behaviours that show up. Even when considering real trees, the class of groups acting on such spaces is vastly more diverse than in the discrete setting: while any simplicial tree admits at most one free vertex transitive action, we will see that there are 2^{2^{\aleph_0}} pairwise non-isomorphic groups which admit a free transitive action on the universal real tree with continuous valence.

29 septembre: Federico Viola (EPFL), "Irreducible finite-index representations of automorphism groups of trees"

Résumé : We study tree automorphism groups and their continuous irreducible representations that preserve a nondegenerate bilinear form of finite index on a Hilbert space. We show that, for groups acting transitively on the boundary of the tree and satisfying Tits' independence property (T), such representations only exist if the bilinear form has index 1.

6 octobre: Yassine Guerch (Caen)
13 octobre: Hanna Oppelmayer (Innsbruck), "Invariant random sub-von Neumann algebras"

Résumé : The notion of IRS (invariant random subgroup) is well-studied in dynamics on groups. We extend this notion to group von Neumann algebras LG, where G is a discrete countable group. We call this concept IRA (invariant random sub-algebra). In particular, we study the case of amenable IRAs, i.e. almost every sub-von Neumann algebra of LG is amenable. This generalises a result of Bader-Duchesne- Lécureux about amenable IRSs. This is joint work with Tattwamasi Amrutam and Yair Hartman. No prior knowledge about von Neumann algebras is assumed.