Aussois 2018

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Scientific program

Two simultaneous actions of big mapping class groups Juliette Bavard - Notes

Mapping class groups of infinite type surfaces, also called "big" mapping class groups, arise naturally in several dynamical contexts, such as two dimensional dynamics, one dimensional complex dynamics, "Artinization" of Thompson groups, etc.
In this mini-course, we will present recent objects and phenomena related to big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). If time allows, we will explain how to use these simultaneous actions to construct non trivial quasimorphisms on subgroups of big mapping class groups. This includes joint work with Alden Walker.

Dimension 1 Z^2 actions Hélène Eynard-Bontemps - Notes

When studying the (nonsingular, C∞) 2-dimensional foliations of 3-manifolds (tangent to the boundary if any), those of T^2 × M , with M = [0, 1] or T^1 , which are transverse to the second factor play an important role. Such a foliation is determined, up to conjugacy, by its holonomy representation, which is an action of Z^2 on M , or, more simply, the data of a pair of commuting diffeomorphisms of M. The study of these objects and their possible deformations requires the understanding of centralizers of diffeomorphisms of M , which will be the main topic of this mini-course. In the case M = T^1, we will only consider diffeomorphisms without fixed points, since the other situation reduces to M = [0, 1]. In this second case, we will focus on diffeomorphisms without interior fixed points (to which one can reduce by subdivision), and actually on germs at 0 and 1, so that we will eventually consider M = R+. We will try to draw a parallel between the two situations (T^1 and R+) as follows :

Applying the theory of Denjoy for the circle and Kopell and Szekeres for R+ , we will see that the centralizer Z_f of an element f of Diff∞(R+) (resp. Diff∞_+(T^1)), with the above restrictions, identifies canonically with a subgroup of R (resp. T^1 , the identification being given in this case by the rotation number). But not every subgroup of R (resp. T^1 ) can be realized this way : there are arithmetic restrictions.
On the one hand, in the circle case, the presence of a diophantine number in this subgroup forces it to be the whole of T^1 . This is a rephrasing of Yoccoz’s famous theorem on the differentiable conjugacy of circle diffeomorphisms to rotations. In this second lecture, we will try to give an idea of the techniques involved in the proof of this theorem.
A maybe not so well-known part of Yoccoz’s thesis consists in constructing circle diffeomorphisms (with a Liouville rotation number α) whose centralizer is reduced to the group of iterates (identified to the dense subgroup αZ/Z of T^1 = R/Z). We will present an analogous result in the case M = R+ based on a construction of Sergeraert and “deformation by conjugation” techniques à la Anosov-Katok.

No knowledge of the above works, even those often considered as classical (Denjoy, Herman, Yoccoz...) is assumed. We will, among other things, present the necessary background on diophantine approximation.

Random walks on mapping class groups, after Mathieu and Sisto Sébastien Gouëzel - Notes

Mathieu and Sisto have introduced a novel method to study random walks on a large class of groups, so-called acylindrically hyperbolic groups (including for instance hyperbolc groups and mapping class groups). They can for instance prove the central limit theorem in this context. I will explain the general setting, several examples, and give the details of several (elementary!) proofs of their results.

Pseudogroups acting on the Cantor set and their full groups Nicolás Matte Bon - Notes

Many well-studied groups arise as groups of homeomorphisms of a compact space that have a simple “local” behaviour, yet jointly generate a complicated group. Examples include Thompson’s groups and generalisations, groups of affine and projective interval exchanges, topological full groups of minimal homeomorphisms, etc. The notion of full group of a pseudogroup formalises this idea.
After reviewing some fundamental results in the theory of full groups of pseudogroup, we will focus on their rigidity properties.

Classification des collection unicellulaire sur une surface de genre g Sane Abdoul Karim

En topologie, les théorèmes de classification sont ceux qu'on recherche le plus; c'est l'objectif majeur; pouvoir classifier les objets topologiques. On peut citer comme exemples le théorème de classification surfaces, le théorème de Lickorish-Wallace sur les 3 variétés obtenu par chirurgie sur la sphère S^3...

Dans cet exposé, on parlera de collections de courbes remplissantes sur une surface de genre g dont le complémentaire est un disque; on définira une operation de chirurgie sur celles ci et on démontrera un théorème de classification modulo cette opération.

Density criterion for isoperiodic leaves Florent Ygouf

The moduli space of abelian differential is endowed with a natural foliation, whose leaves are locally described by varying the relative periods, while fixing the absolute ones. The study of this foliation, called the isoperiodic foliation, has been initiated in the 90’s by Kontsevich and Eskin, and later by Calta and McMullen. It is since a important tool in Teichmüler dynamics. However, it is only very recently that results on the dynamics of its leaves have been obtained. In 2015 Hooper and Weiss gave the first examples of dense leaves (the ones through the Arnoux-Yoccoz surfaces). Shortly after, Calsamiglia, Deroin, Francaviglia obtaied a Ratner-like classification of the minimal sets in the principal stratum. Simultaneously, Hamenstädt obtained the ergodicity for the Masur-Veech measure in the same setting.

In this talk, after carefully defining the isoperiodic foliation, I will give a criterion of density for leaves of the isoperiodic in rank 1 affine manifolds. I will explain how this criterion can be used to discover new dense leaves and new non-trivial minimal sets.

Fried surgeries on Anosov flows Mario Shannon

The two most basic families of examples of Anosov flows in dimension three are the geodesic flows of hyperbolic surfaces and the suspension flows generated by Anosov diffeomorphisms of the 2-dimensional torus, both fam- ilies also called the algebraic Anosov flows. Non-algebraic Anosov flows where constructed for the first time in 1979 and 1980 (Franks-Williams for the non-transitive case and Handel-Thurston for the transitive one) by the use of surgery procedures, which consists basically in cut a 3-manifold pos- sessing some hyperbolic flow along incompressible tori and then glue them back in a non-trivial way. Around the same time S. Goodman described a surgery procedure where, given an Anosov flow and a periodic orbit, it al- lows to perform a Dehn surgery in a tubular neighbourhood of this orbit and obtain a new manifold with a Anosov flow. As a more sophisticated tool, in his 1983 work D. Fried proposed another procedure to perform a Dehn surgery along a periodic orbit, which can be interpreted a an infinitesimal version of the Goodman surgery. The main advantage of the Fried’s proce- dure with respect to the Goodman surgery is that we can keep track of the topology of stable/unstable foliations of the original flow during this surgery. The problem is that, although the new flow preserves two 1-dimensional fo- liations for which the distance inside the leaves are contracted/expanded by the positive/negative action of the flow, it is not obvious that this flow is Anosov, the main problem being related to check the differentiability and the exponential rates of contraction/expansion. (The new flow is a so called topologically Anosov flow.)

Our aim is to show the following:
Given an Anosov flow in a 3-manifold, a Fried surgery along a periodic orbit produces a flow which is topologically equivalent to a (true) Anosov flow.

Schedule

Date	Time	Event
3 Monday December, 2018	9:30 - 10:30	Juliette Bavard
	11:00 - 12:00	Nicolás Matte Bon
	14:00 - 15:00	Hélène Eynard-Bontemps
	15:30 - 16:30	Mario Shannon
4 Tuesday December, 2018	9:30 - 10:30	Sébastien Gouëzel
	11:00 - 12:00	Juliette Bavard
	14:00 - 15:00	Nicolás Matte Bon
	15:30 - 16:30	Olivier Glorieux
5 Wednesday December, 2018	9:30 - 10:30	Hélène Eynard-Bontemps
	11:00 - 12:00	Sébastien Gouëzel
	14:00 - 20:00	Free afternoon
	20:00 -	Fondue evening
6 Thursday December, 2018	9:30 - 10:30	Nicolás Matte Bon
	11:00 - 12:00	Hélène Eynard-Bontemps
	14:00 - 15:00	Juliette Bavard
	15:30 - 16:30	Abdoul Karim Sane
7 Friday December, 2018	9:30 - 10:30	Sébastien Gouëzel
7 Friday December, 2018	11:00 - 12:00	Florent Ygouf

Come to the new edition of Aussois conference.

From 3 to 7 December 2018 !

Scientific program

Two simultaneous actions of big mapping class groups Juliette Bavard - Notes

Dimension 1 Z^2 actions Hélène Eynard-Bontemps - Notes

Random walks on mapping class groups, after Mathieu and Sisto Sébastien Gouëzel - Notes

Pseudogroups acting on the Cantor set and their full groups Nicolás Matte Bon - Notes

Classification des collection unicellulaire sur une surface de genre g Sane Abdoul Karim

Density criterion for isoperiodic leaves Florent Ygouf

Fried surgeries on Anosov flows Mario Shannon

Schedule

Practical information

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Participants

Thank you !

This conference was supported by Fondation Louis D.