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.
- 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.
- 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.
- 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.