Habilitation à Diriger des Recherches

Voici mon mémoire d'HDR, intitulé Polish groups and non free actions in the discrete or measurable context, ainsi que les slides de la soutenance, effectuée le 14 mars 2024.

A morning in groups and dynamics

Following the HDR defense, we had A morning in groups and dynamics on March 15 2024, here are the titles and abstracts.

• 9:00-10:00 : The interplay between rigid and strongly proximal actions of some C*-simple groups with applications to non-abelian factors (Yair Glasner, Ben Gurion University of the Negev).
  We study intermediate C*-algebras of the form C*_r(Γ) < 𝒜 < C(X) ⋊ Γ, where Γ ⤳ X is a given minimal action of a countable discrete group Γ on a compact space X. Every Γ-factor of the given topological dynamical system X → Y gives rise to an intermediate algebra of the form 𝒜 = C(Y) ⋊ Γ, and by analogy, we may think of more general factors as representing "non-abelian" factors. Let us call the dynamical system "reflecting" if the only intermediate algebras come from dynamical factors. A topological dynamical system Γ ⤳ X is called rigid if the corresponding image ρ(Γ) < Homeo(X) is non-discrete as a subgroup of the Polish group of homeomorphisms of X. In particular, any equicontinuous action of an infinite group is rigid. We show that a large family of groups, including all hyperbolic groups with no finite normal subgroups, have the property that every rigid topological dynamical system is reflecting. This is joint work with Tattwamasi Amrutam and Eli Glasner.
• 10:00-10:30 : Coffee break.
• 10:30-11:30 : The space of transitive Cantor actions of a countable group (Julien Melleray, Université Lyon 1).
  We are interested in the space of transitive actions of a countable group on the Cantor space; in particular we would like to know when there is a dense conjugacy class in this space, and understand which actions can be approximated by transitive actions (i.e. describe the closure of this space). These questions admit an easy answer for groups which are not finitely generated, and it turns out that the situation is much more complicated for finitely generated groups. I will present some partial answers, mostly for nilpotent groups and for free groups. This is part of joint work with M. Doucha (Prague) and T. Tsankov (Lyon).
• 11:30-12:30 : Growth of homology in towers of finite coverings (Quand Nicolas monte dans les tours) (Damien Gaboriau, ENS de Lyon).
  I intend to report on some joint work with Nicolas Bergeron studying the way the homology grows with respect to the index when taking sequences of successive finite coverings of compact spaces. We shall discuss the Betti numbers along non Galois towers (non-regular Lueck approximation theorem) and the growth of the homology torsion (this is also joint with Miklós Abért and Mikolaj Fraczyk).