Matthieu JOSEPH

Preprints:

  • • New On dissociated infinite permutation groups (with R. Barritault and C. Jahel)

    ArXiv

    The goal of this paper is threefold. First, we describe the notion of dissociation for closed subgroups of the group of permutations on a countably infinite set and explain its numerous consequences on unitary representations (classification of unitary representations, Property (T), Howe-Moore property, etc.) and on ergodic actions (non-existence of type III non-singular actions, Stabilizer rigidity, etc.). Some of the results presented here are new, others were proved in different contexts. Second, we introduce a new method to prove dissociation. It is based on a reinforcement of the classical notion of strong amalgamation, where we allow to amalgamate over countable sets. Third, we apply this technique of amalgamation to provide new examples of dissociated closed permutation groups, including isometry groups of some metrically homogeneous graphs, automorphism groups of diversities, and more.
  • • Unitary representations of the isometry groups of Urysohn spaces (with R. Barritault and C. Jahel)

    ArXiv

    We obtain a complete classification of the continuous unitary representations of the isometry group of the rational Urysohn space ℚ𝕌. As a consequence, we show that Isom(ℚ𝕌) has property (T). We also derive several ergodic theoretic consequences from this classification: (i) every probability measure-preserving action of Isom(ℚ𝕌) is either essentially free or essentially transitive, (ii) every ergodic Isom(ℚ𝕌)-invariant probability measure on [0,1]ℚ𝕌 is a product measure. We obtain the same results for isometry groups of variations of ℚ𝕌, such as the rational Urysohn sphere ℚ𝕌1, the integral Urysohn space ℤ𝕌, etc.
  • • Stabilizers for ergodic actions and invariant random expansions of non-archimedean Polish groups (with C. Jahel)

    accepted in Int. Math. Res. Not.
    ArXiv

    Let G be a closed permutation group on a countably infinite set Ω, which acts transitively but not highly transitively. If G is oligomorphic, has no algebraicity and weakly eliminates imaginaries, we prove that any p.m.p. ergodic action G ↷ (X, µ) is either essentially free or essentially transitive. A key notion that we develop in our approach is that of invariant random expansions, which are G-invariant probability measures on the space of expansions of the canonical (model theoretic) structure associated with G. We also initiate the study of invariant random subgroups for Polish groups and prove that – although the result for p.m.p. ergodic actions fails for the group Sym(Ω) of all permutation of Ω – any ergodic invariant random subgroup of Sym(Ω) is essentially transitive.
  • • Isometric orbit equivalence for probability-measure preserving actions

    ArXiv

    We introduce the notion of isometric orbit equivalence for probability-measure preserving (pmp for short) actions . This notion asks the Schreier graphings defined by the actions of the groups to be isomorphic. First, we prove that the pmp actions of a group whose Cayley graph has a discret automorphisms group are rigid up to isometric orbit equivalence. In a second time, we introduct a method which enables us to prove that several groups admit pmp actions which are isometric orbit equivalent but not conjugate. This includes for instance free groups of finite rank with a free generating set.

Publications:

  • • Amenable wreath products with non almost finite actions of mean dimension zero

    Trans. Amer. Math. Soc., 377, No. 2, 1321-1333 (2024)
    Published version | ArXiv | Video

    Almost finiteness was introduced in the seminal work of Kerr as an dynamical analogue of Z-stability in the Toms-Winter conjecture. In this article, we provide the first examples of minimal, topologically free actions of amenable groups that have mean dimension zero but are not almost finite. More precisely, we prove that there exists an infinite family of amenable wreath products that admit topologically free, minimal profinite actions on the Cantor space which fail to be almost finite. Furthermore, these actions have dynamical comparison. This intriguing new phenomenon shows that Kerr's dynamical analogue of Toms-Winter conjecture fails for minimal, topologically free actions of amenable groups. The notion of allosteric group holds a significant position in our study. A group is allosteric if it admits a minimal action on a compact space with an invariant ergodic measure that is topologically free but not essentially free. We study allostery of wreath products and provide the first examples of allosteric amenable groups.
  • • Belinskaya's Theorem is optimal (with A. Carderi, F. Le Maître and R. Tessera)

    Fund. Math., 263 (2023), n°1, 51-90.
    Published version | ArXiv | Video (by F. Le Maître)

    Belinskaya's theorem states that given an ergodic measure-preserving transformation, any other transformation with the same orbits and an L1 cocycle must be flip-conjugate to it. Our main result shows that this theorem is optimal: for all p<1, the integrability condition on the cocycle cannot be relaxed to being in Lp. This also allows us to answer a question of Kerr and Li: for ergodic measure-preserving transformations, Shannon orbit equivalence doesn't boil down to flip-conjugacy.
  • • Continuum of allosteric actions for non-amenable surface groups

    Erg. Th. Dyn. Syst., 44, No. 6, 1581-1596 (2024).
    Published version | ArXiv

    Let Σ be a closed surface other than the sphere, the torus, the projective plane or the Klein bottle. We construct a continuum of p.m.p. ergodic minimal profinite actions for the fundamental group of Σ, that are topologically free but not essentially free, a property that we call allostery. Moreover, the IRS's we obtain are pairwise distinct.
  • • Products of snowflaked Euclidean lines are not minimal for looking down (with T.Rajala)

    Anal. Geom. Metr. Spaces, 5 (2017), no. 1, 78-97.
    Published version | ArXiv

    We study a notion of self-similarity for metric spaces called BPI (Big Pieces of Itself). In order to classify all the different BPI geometries, David and Semmes introduced the notion of "looking down". In this article, we prove that the product of finitely many euclidean lines, all equipped with the distance (x,y)→|x-y|p for some p<1, is not minimal for looking down.

PhD Thesis:

  • Topological and measurable dynamics: allostery, quantitative orbit equivalence

    Under the supervision of Damien Gaboriau.

    TEL | Pdf version

    This PhD thesis lies at the interface between topological dynamics and measurable dynamics. First, I study the notion of allosteric actions. These actions are generically free in the sense of the topology but not generically free in the sense of the measure. This surprising behavior highlights the differences between invariant random subgroups and uniformly recurrent subgroups. The nascent theory of quantitative orbit equivalence is the second topic of this thesis. This is a strengthening of orbit equivalence, which aims to understand how metric structures on the orbits of the actions can be distorted. A large part of my work gravitates around one of the founding result of this theory: Belinskaya’s theorem.