Groupe de travail "théorie ergodique"

Archives du GdT de théorie ergodique

2024-2025

This year, we had quite a diversity of topics. The BBB server featuring recordings of the talks was sadly shut and we had to move to this one at the end of the year. In order to be less dependent of BBB servers for recording archives, all recordings were converted to mp4 files and put on this personnal webserver (they are too heavy for this website). You will find links to these mp4 files below.

18 avril :The existence of ergodic hyperfinite subgraphs III (Anush Tserunyan). Here are the recording and the notes from the talk. Abstract
In this series of talks, I will discuss my proof (2018–2022) of the existence of ergodic hyperfinite subgraphs in any locally countable measured ergodic graph (not necessarily pmp). This result yields a pointwise ergodic theorem for measured graphs and serves as a starting point for several constructions in measured group theory for treeable equivalence relations, since it provides an ergodic hyperfinite free factor.
In the pmp setting, this theorem was first proved by R. Tucker-Drob, who ingeniously used a major result of Hutchcroft and Nachmias from percolation theory as a black box. My proof differs even for pmp graphs and extends to the non-pmp setting. It relies only on basic tools from descriptive set theory for working with measured locally countable graphs and exploiting non-amenability.
11 avril : The existence of ergodic hyperfinite subgraphs II (Anush Tserunyan). Here are the recording and the notes from the talk.
4 avril : The existence of ergodic hyperfinite subgraphs I (Anush Tserunyan). Here are the recording and the notes from the talk.
21 mars : Exponential mixing and equipartition along spheres for stationary processes on regular trees (Elias Zimmermann). The talk was on-site and thus not recorded. Abstract
We consider generalizations of Shannon's asymptotic equipartition property and its strengthenings due to McMillan and Breiman for stationary processes over Cayley graphs. While for graphs arising from amenable groups equipartition is well understood due to work of Ornstein, Weiss and Lindenstrauss, only few results could be established so far for Cayley graphs with a non-amenable geometry such as regular trees. In this talk I will present a method for proving equipartition results for stationary processes over regular trees under the assumption of a quantitative mixing condition by studying actions of suitable amenable groups on the boundary of the tree. I will also discuss examples of appropriately mixing processes arising from the theory of Gibbs measures.
7 mars : A quasi-isometric classification of permutational wreath products II (Vincent Dumoncel). Here is the recording of the talk. Abstract
It is in general a hard problem to determine whether two given finitely generated groups are quasi-isometric, even among some "well behaved" classes. One such class, which has been the subject of intensive research in group theory, is the one of wreath products, as they often exhibit unexpected and interesting behaviours.
The classification up to quasi-isometry of lamplighters over Z goes back to 2013, and in a recent work (2021), Genevois and Tessera extended it to all lamplighters over finitely presented one-ended groups. This raises the question of also classifying their permutational variants. In this context, strong rigidity phenomenon as the ones observed for standard wreath products do not hold, whence the need of another approach. After having introduced and discussed several re-inforcements of quasi-isometries, I will sketch the main guidelines of the proof of a quasi-isometric classification of some permutational wreath products, that covers a number of classical cases. If time permits, we will also discuss some applications and open problems.
21 février : A quasi-isometric classification of permutational wreath products I (Vincent Dumoncel). Here are the notes and the recording of the talk.
14 février : On pmp Boolean actions of the group of permutations of the integers (Matthieu Joseph). Here are the notes and the recording of the talk. Abstract
I will discuss several aspects of pmp Boolean actions for the Polish group Sym(N), such as classification of Bernoulli shifts, or else a generalization of Kolmogorov and Hewitt-Savage 0-1 laws. This is based on joint works with R. Barritault, C. Jahel and E. Roy.
7 février : Perfect kernel of generalized Baumslag-Solitar groups (Sasha Bontemps). Here are the notes from the talk and its recording. Abstract
Endowed with the Chabauty topology, the space of subgroups Sub(G) of any infinite countable group G is a closed subset of the Cantor space, on which G acts by conjugation. The perfect kernel of G is the largest closed subset of Sub(G) without isolated points. It is invariant under conjugation.
In this talk, we study the space of subgroups of generalized Baumslag-Solitar groups, i.e. groups acting cocompactly on an oriented tree with infinite cyclic edge and vertex stabilizers. As a result of Bass-Serre theory, these are exactly the finite iterations of HNN-extensions and amalgamated free products of Z over Z starting from Z. Our results generalize those obtained by Alessandro Carderi, Damien Gaboriau, François Le Maître and Yves Stalder obtained for Baumslag-Solitar groups in 2022. Given a non amenable GBS group G acting on its (reduced) Bass-Serre tree T, we show the existence of an infinite countable G-invariant partition of the perfect kernel such that:
- one piece of the decomposition is closed, and all the other ones are open (and closed iff G is virtually the direct product of a free group with Z);
- there exists a dense orbit in each of these pieces.
The proof works by "cutting and pasting" quotients of T by subgroups of G. This partition relies on a generalization of the "phenotype" defined by the aforementionned authors. This is a function from the set of subgroups of G to N that is invariant under conjugation and that depends on the indices of the quotient graph G\T and on the indices of the edge stabilizers into the vertex stabilizers of T.
31 janvier : Ergodic theory of non-Archimedean Polish groups (Colin Jahel). Voici les notes de l'exposé ainsi que l'enregistrement.
24 janvier : Odomutants and flexibility results for quantitative orbit equivalence II (Corentin Correia). Here are the notes from the talk, the recording and the detailed notes. Abstract
Let us go back to the study of our favorite group Z! After recalling the main results on quantitative orbit equivalence for probability measure-preserving Z-actions, I will explain my recent work on systems that I call "odomutants". These systems are built by distorting the orbits of odometers. They have various nice properties which enable us to obtain three flexibility results in quantitative OE.
The first one is the non preservation of Kakutani equivalence under L^<1/2 orbit equivalence, thanks to a construction of Feldman (which is actually the starting point of my work). The second one is the optimality of Kerr and Li's theorem, stating that measure-theoretical entropy is preserved under log-integrable orbit equivalence. It turns out that the proof uses a more topological aspect of OE, called strong OE, and we can draw a parallel with a famous theorem of Boyle and Handelman. Finally, I proved that for every sublinear map phi, phi-integrable orbit equivalence to any odometer does not imply flip-conjugacy, thus extending a theorem by Carderi, Joseph, Le Maître and Tessera.
17 janvier : Odomutants and flexibility results for quantitative orbit equivalence I (Corentin Correia). Here are the notes from the talk, its recording as well as the detailed notes.
10 janvier : A generalization of Dye's reconstruction theorem by Fremlin (Fabien Hoareau). Here are the notes from the talk, therecording and finally the detailed notes. Abstract
Dye's reconstruction theorem states that any abstract isomorphism between two ergodic full groups on a standard probability space is actually a conjugation by some measure-preserving bijection. This is useful for instance for proving that full groups are complete invariants of orbit equivalence. When working with infinite measure or with non-singular bijections however, we need a more general version of the theorem, which was later proved by Fremlin in the language of Boolean algebras. I will give some details about his proof, which relies on an algebraic description of the involutions in the full group, and will work specifically in the context of measure algebras.
6 décembre : Tautness and measure equivalence rigidity (Sasha Gurieva). Here are the notes from the talks and the recording. Abstract
A countable group G is said to be ME-rigid if its measure equivalence class is as small as possible, i.e. it contains only groups that are virtually isomorphic to G (up to finite kernels).
In 1999, Furman shows that higher rank lattices are ME-rigid by applying Zimmer's superrigidity of cocycles. His strategy of proof turned out to be quite general, and was later developed by Monod-Shalom, Kida and Bader-Furman-Sauer. In their work, proving ME-rigidity of their chosen group G is split into two distinct steps :
1. Exploiting the specific geometry of G to obtain a form of rigidity of self-ME-couplings of G, called tautness.
2. Showing that any taut and ICC group is indeed ME-rigid.
I will describe some techniques used for ME-rigidity, like this second step, common to all four of the latter articles.
29 novembre : On growth of cocycles for isometric representations on Banach spaces III (Juan Paucar). Here are the notes from the talk and the recording. Abstract
It is known by the Delorme-Guichardet theorem that for σ-compact locally compact groups property (T) coincides with property FH, that is, the property that every continuous isometric action on a Hilbert space has a fixed point. Another rephrasing of this property is the fact that for any unitary representation all 1-cocycles have to be bounded. Inspired by the work of V.Lafforgue, we will explore the asymptotics of how unbounded can the 1-cocycles be in the case of a group without property FLp, an Lp-analogue of property FH, more precisely we will study the behaviour as n grows to infinity of sup|g|≤ n ||b(g)|| and ||b||Lp(μ*n) where μ is an appropriate probability in G.
22 novembre : On growth of cocycles for isometric representations on Banach spaces II (Juan Paucar). Here are the notes from the talk and the recording.
15 novembre : On growth of cocycles for isometric representations on Banach spaces I (Juan Paucar). Here are the notes from the talk and the recording.
8 novembre : Around the commensurating full group III. (Antoine Derimay). Here are the notes from the talk and therecording. Abstract
Deciding whether two automorphisms of a standard Borel space are conjugate is a historically challenging problem, creating numerous new methods in an attempt at solving it. Recently however, it has been shown by Foreman-Rudolph-Weiss that this equivalence relation is not Borel, meaning we have little chance of understanding it, ever. A half-century before this result was known, Belinskaya showed that if two ergodic pmp automorphisms have the same orbits, and one has an integrable cocycle with respect to the other, then they are flip-conjugate. This result encouraged Le Maître in 2018 to introduce the L1 full group of a pop automorphism T, defined as the group of automorphisms preserving the T-orbits such that their cocycle is integrable. This group has proven to have a number of nice properties: it is Polish, its quasi-isometry type is known, it is an algebraic invariant of flip-conjugacy etc. However, those results can only hold when T is pmp, which is a restricted setting. A solution to this problem, inspired by an alternative proof of Belinskaya’s theorem by Katznelson is to consider instead the commensurating full group. In my talk, I will define the important notions of this area of research, before defining the commensurating full group and proving that many of the properties of the L1 full group still hold, and, if time allows it, I will also give some extra properties that the L1 full group doesn’t have.
25 octobre : Around the commensurating full group II (Antoine Derimay). Here are the notes from the talk (truncated because of technical difficulties) as well as the recording (which starts roughly 1 minute late because of additional technical difficulties!).
11 octobre : Around the commensurating full group I (Antoine Derimay). Here are the notes from the talk and the recording.
4 octobre : Sur les représentations unitaires et la théorie ergodique des groupes polonais non archimédiens III (Matthieu Joseph). Voici les notes de l'exposé ainsi que l'enregistrement. Abstract
Soit G un sous-groupe fermé du groupe Sym(Ω) des permutations d’un ensemble dénombrable Ω. On s’intéresse aux trois problèmes suivants :
- Classifier les représentations unitaires de G,
- Classifier les mesures de probabilités ergodiques G-invariantes sur [0,1]^Ω,
- Classifier les sous-groupes aléatoires invariants de G.
Dans un premier temps, on expliquera comment une propriété abstraite — appelée dissociation — définie au niveau des représentations unitaires, permet de résoudre ces trois problèmes. On illustrera cette propriété dans un second temps pour résoudre concrètement ces trois problèmes lorsque G est le groupe des isométries de l’espace d’Urysohn rationnel.
Ceci est le fruit d’un premier travail avec C. Jahel et d’un autre avec R. Barritault et C. Jahel.
27 septembre : Sur les représentations unitaires et la théorie ergodique des groupes polonais non archimédiens II (Matthieu Joseph). Voici les notes de l'exposé ainsi que l'enregistrement.
20 septembre : Sur les représentations unitaires et la théorie ergodique des groupes polonais non archimédiens I (Matthieu Joseph). Voici les notes de l'exposé ainsi que l'enregistrement.

2023-2024

This year was mainly devoted to restricted orbit equivalence. The BBB server featuring recordings of the talks was sadly shut. The videos have been extracted before that happened but are now hosted on a personnal webserver since they are too heavy to be hosted directly here.

April 2: Actions minimales sur le Cantor sans partition de Kakutani-Rokhlin. (Matthieu Joseph).
December 4 : Exhasutive OE and sizes, after Heicklen (François Le Maître). Here are the recording and the notes from the lecture.
November 21 : Introduction to Kakutani equivalence II (Corentin Correia). Here are the recording and the notes from the lecture.
November 7 : Introduction to Kakutani equivalence (Corentin Correia). Here are the recording, the notes from the lecture and Corentin's more detailed typeset notes, featuring very nice pictures!
October 24 : Exhaustive orbit equivalence as a restricted orbit equivalence, after Heicklen (François Le Maître). Here are the notes (the recording is not available since we forgot to turn it on).
October 10 : Restricted orbit equivalence from the metric viewpoint (François Le Maître). Here are the recording and the notes.
September 26 : Mixing for dyadic equivalence, after Hasfura-Buenaga (Matthieu Joseph). Here is the recording.
24 avril : Entropie et théorème de Kolmogorov-Sinai (Matthieu Joseph). Voici les notes d'exposé et la vidéo. Enfin voici un scan des notes préparatoires de Matthieu.
9 mai : Le théorème des générateurs de Rokhlin (Matthieu Joseph). Voici les notes d'exposé, un scan des notes préparatoires, et la vidéo.
16 mai: Le théorème d'équivalence orbitale de Dye (François Le Maître). Voici les notes d'exposé, et la vidéo (malheureusement l'enregistrement a été déclenché après 20 minutes).
30 mai : Belinskaya's theorem (Fabien Hoareau). Voici les notes d'exposé, les notes préparatoires ainsi que la vidéo.
13 juin : Invariance de l'entropie par équivalence orbitale de Shannon pour les actions p.m.p. de Z (Corentin Correia). Voici les notes d'exposé, les notes détaillées ainsi que la vidéo.
20 juin :Équivalences orbitales pour les groupes localement finis, d'après Vershik et Stepin (François Le Maître). Voici les notes d'exposé, some detailed notes on Stepin's Theorem 1, ainsi que la vidéo.
27 juin : Équivalence orbitale bornée pour les actions de Zⁿ, d'après Fieldsteel et Friedman (Matthieu Joseph) Voici les notes d'exposé ainsi que la vidéo.

2019-2020

Ce semestre, on prévoit une série d'exposés sur les sous-algèbres maximales des algèbres de von Neumann de groupes. On étudiera les trois articles suivants : l'article de Yongle Jiang : Maximal von Neumann subalgebras arising from maximal subgroups et les articles de Rémi Boutonnet et Alessandro Carderi : Maximal amenable subalgebras of von Neumann algebras associated with hyperbolic groups et Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups.

21 février : Sous-algèbres maximales moyennables dans les algèbres de von Neumann des groupes hyperboliques, d'après Boutonnet Carderi. (Frank Taipe)
6 février : Groupes hyperboliques et leur dynamique au bord II (Kostya Slutsky)
30 janvier : Groupes hyperboliques et leur dynamique au bord (Tomás Ibarlucı́a)
23 janvier : Critère de Boutonnet-Carderi pour qu'une algèbre de groupe soit maximale Gamma (Simeng Wang)
17 janvier : Actions 4-transitives et sous-algèbres maximales, d'après Jiang (François Le Maître)
25 octobre : Bord de Poisson de groupes polonais moyennables, d'après Schneider-Thom (François Le Maître)
18 octobre : Bord de Poisson pour des groupes dénombrables (Adrien Boyer)
27 septembre : Constructing the Poisson boundary for general measured topological groups (Andrew Zucker)
20 septembre : Følner sets for amenable Polish groups (after Schneider and Thom) (Todor Tsankov)
13 septembre : Hyperfinite witnesses for Z^d and R^d actions (by Gao and Jackson) (Kostya Slutsky)

2018-2019

vendredi 15 février: Random Walks, The Poisson Boundary, and the ICC property (Joshua Frisch)
vendredi 30 novembre: Moyennabilité des groupes pleins topologiques d'homéomorphismes minimaux du Cantor (François Le Maître)
vendredi 16 novembre: Récurrence de marches aléatoires et moyennabilité extensive (Pierre Giraud)
vendredi 9 novembre: Moyennabilité extensive (Andy Zucker)
vendredi 26 octobre: Théorème de Kesten (Arman Darbinyan)
vendredi 12 octobre: Critère de Nash-Williams (Colin Jahel)
vendredi 5 octobre: Systèmes distaux et ergodicité forte (Tomás Ibarlucía)
Vendredi 28 septembre: Lemme de Rohlin pour les actions pmp hyperfinies (Pierre Giraud)

2017-2018 (équivalence orbitale)

Vendredi 13 avril: Le théorème de Dye, d'après Katznelson-Weiss (François Le Maître)
Vendredi 16 mars: Le théorème d'Häggström-Peres sur la phase d'unicité en percolation (Colin Jahel)
Vendredi 24 février: Une caractérisation de la propriété (T) pour les groupes dénombrables (d'après Robertson-Steger) (François Le Maître)
Vendredi 12 janvier: Trou spectral pour les relations d'équivalences fortement ergodiques et relations d'équivalences stables (Amine Marrakchi)
Vendredi 7 décembre: Résultats généraux pour la percolation sur des graphes de Cayley (Colin Jahel)
Vendredi 17 novembre: Théorème d'Ornstein-Weiss (Todor Tsankov)
Vendredi 3 novembre: Coût des actions libres de groupes libres (François Le Maître)
Vendredi 6 octobre: Introduction à l'équivalence orbitale (François Le Maître)

Références

Orbit Equivalence and Measured Group Theory, D. Gaboriau, Proceedings of the ICM (Hyderabad, India, 2010), Vol. III, Hindustan Book Agency (2010).
Topics in orbit equivalence, A. S. Kechris and B. D. Miller, Springer. (2004)

2016-2017 (entropie)

Jeudi 26 janvier: conjugaison de transformations préservant la mesure et théorème de Kolmogorov-Sinai (François). Some notes on the Kolmogorov-Sinai theorem.
Jeudi 23 février: le théorème de Shannon-McMillan-Breiman (Tomás).
Jeudi 16 mars: A zero-error Rokhlin lemma for amenable groups (after Conley, Jackson, Kerr, Marks, Seward, and Tucker-Drob) (Todor)
Jeudi 23 mars: Quasi-tilings for amenable groups (François)
Mercredi 3 mai: f-entropie des groupes libres (Andreas)
Jeudi 18 mai: le théorème de Stepin (Tomás).
Mercredi 24 mai Le théorème du générateur de Rohlin (François ).
(date à préciser) Entropie des groupes moyennables.

Références

L. Bowen, « A measure-conjugacy invariant for free group actions », Annals of Mathematics, vol. 171, n^o 2, p. 1387‑1400, mars 2010.
T. Downarowicz, Entropy in dynamical systems, vol. 18. Cambridge University Press, Cambridge, 2011.
E. Glasner, Ergodic Theory via Joinings, vol. 101. Providence, Rhode Island: American Mathematical Society, 2003.