Description: The school, related to the international research project Dynamical Group Theory, funded by MathAMSUD, aims at introducing young researchers
to different modern perspectives of groups of transformations and their actions.
The related event Big Mapping Class Groups and Diffeomorphism Groups will take place at CIRM in Marseille the following week (October 10-14, 2022).
Organizers: Joaquín Brum (Universidad de la República), Nicolás Matte Bon (CNRS, Université de Lyon 1), Cristóbal Rivas (Universidad de Chile), Michele Triestino (Université de Bourgogne)
Dates: October 3-7, 2022
Mini-courses
Abstract. Let \(M\) be a compact interval or a circle. We study algebraic properties of \(\mathrm{Diff}(M;r)\), the group of \(C^r\) diffeomorphisms of \(M\). In particular, we focus on the following question: which finitely generated groups arise as subgroups of \(\mathrm{Diff}(M;r)\)? We survey classical theories of Hölder, Denjoy, Kopell and Thurston regarding this matter, and give a detailed dynamical account of a finitely generated subgroup of \(\mathrm{Diff}(M;r)\) that does not admit any injective homomorphism into \(\mathrm{Diff}(M;s)\) for all \(s>r\). (joint with Thomas Koberda).
Abstract. In this mini-course we will focus on the Cremona group, the group of birational transformations of the projective plane. After a gentle introduction, we will construct in a first time its action on a infinite-dimensional hyperboloid and in a second time its action on a \(\mathrm{CAT}(0)\)-cube complex. In each case, we will obtain group properties and dynamical results of the Cremona group.
Abstract. In this mini-course, I will first explain why the group of homeomorphisms of a closed manifold isotopic to the identity is simple and why the same holds for most groups of diffeomorphisms. Then, I will talk about works by Burago, Ivanov and Polterovitch and Tsuboi about the commutator length on those groups. If time allows, I will also talk about a recent work by Bowden, Hensel and Webb which deals with the very special case of surfaces.
Research talks
Abstract. A laminar group is a subgroup of orientation preserving circle homeomorphisms preserving circle laminations. Laminar group theory is motivated by Thurston's universal circle theorem. The theorem says that a tautly foliated three manifold group acts on the (universal) circle preserving a pair of circle laminations. Laminar group theory studies the converse of this theorem. In this talk, I will introduce some basic notions and recent progress. This work is joint with Hyungryul Baik and Hongtaek Jung.
Abstract. An (infinite order) element of a group is distorted if the word-length of its iterates grows sublinearly, in some finitely generated subgroup. A general question is to know which diffeomorphisms are distorted in the corresponding group of diffeomorphisms. In particular, we will see that there are diffeomorphisms of the interval which are distorted in \(\mathrm{Diff}^1(I)\) but undistorted in \(\mathrm{Diff}^2(I)\). If time allows, we will also discuss the (open) problem for germs at the origin.
PROGRAM (go to this page)
List of participants (go to this page)
Venue: The school will take place at the Institut de Mathématiques de Bourgogne, Dijon (France).
Recommended hotel: Participants will preferably stay at Hotel Kyriad Dijon Mirande, at 5 minutes walk from the Math department.