Groupes d'homéomorphismes de variétés

Groups of homeomorphisms of manifold

ANR PRC Gromeov ANR-19-CE40-0007

Description

The study of group actions on manifolds is an extension of classical dynamical systems. It is a combination of algebra and dynamics: the algebraic structure makes the dynamics rigid, while the dynamics provide algebraic information. The complexity of the problem leaves us with plenty of unsolved questions even for actions in dimension 1 and 2. If in dimension 1 we are in a good position for reaching some sort of classification, this is certainly not the case in dimension 2. The keyword for the project could be "interaction": between group theory and dynamical systems, topology and geometry; between dimension 1 and 2. This project gathers experts of the domain, with different skills. Young researchers represent a large proportion of the members. This project has 3 principal research tracks.

1. For actions in dimension 1, we want to make advances towards well-known problems, such as Zimmer's program, Hector-Ghys-Sullivan conjectures (relationship between ergodicity and minimality for $$C^2$$ codimension-one foliations), study of finitely generated simple groups acting on the line, random dynamics with applications to Schrodinger operators. We will count on the several partial results have been obtained in the recent years by members of the consortium.
2. For actions in dimension 2, there are only sparse results, and a first objective of the project is to seriously review what is known (with a monograph). We plan to make a more systematic use of deep results in topological dynamics to develop new tools for group actions. An other promising approach is by studying minimal invariant subsets, which provide natural representations to big mapping class groups, which constitute a very active topic today, at the interface between low-dimensional topology, geometric group theory and hyperbolic geometry. Finally, we plan to investigate the large scale geometry of the group of all homeomorphisms of a manifold, following recent ideas introduced by Mann and Rosendal. The members involved into this part of the project will benefit from the interaction with the other members which are experts in one-dimensional action.
3. We will also study a particular class of group actions, which comes from classical dynamical systems. Given an Anosov flow $$X$$ on a compact 3-manifold $$M$$, by an old result of Fenley the fundamental group of $$M$$ acts on a bifoliated plane, and by Calegari-Dunfield (following an idea of Thurston), it also act on a topological circle. Understanding the dynamics of these actions gives an insight into the dynamics of the Anosov flow $$X$$. One of our approaches will be through the description of what natural surgeries on Anosov flows affect these actions. Among the members of the consortium, Barbot and Bonatti are among the best experts on this problem.
We plan to organize several meetings, in particular a thematic trimester: the subject of group actions on manifold has received great attention in these last years and this will be the opportunity to discuss the progresses, inviting the best international experts in France. A particular attention will be given to organize events for students and young researchers. At the end of this project, a large conference will be organized, and this will be the occasion to share the results obtained during the four years of the project.

Forthcoming and recent events

Partially related to our project:

Recent scientific production

Preprints:
• A. Gordenko - Random dynamical systems on a real line, arXiv:2009.14686
• A. Gordenko - Limit shapes of large skew Young tableaux and a modification of the TASEP process, arXiv:2009.10480
• C. Hirsch, M. Holmes, V. Kleptsyn - WARM percolation on a regular tree in the strong reinforcement regime, arXiv:2009.07682
• V. Gorin, V. Kleptsyn - Universal objects of the infinite beta random matrix theory, arXiv:2009.02006
• C. Bonatti, I. Iakovoglou - Anosov flows on 3-manifolds: the surgeries and the foliations, arXiv:2007.11518
• D. Malicet - Lyapunov exponent of random dynamical systems on the circle, arXiv:2006.15397
• A. Le Boudec, N. Matte Bon - A commutator lemma for confined subgroups and applications to groups acting on rooted trees, arXiv:2006.08677