arXiv subjects

François Le Maître

Publications and source records attributed to François Le Maître.

At least 19 recordsLinked to original sources

arXiv2026

Classification of non-free p.m.p. boolean actions of ergodic full groups and applications

We extend Dye's reconstruction theorem, which classifies isomorphisms between full groups, to a classification of homomorphisms between full groups. For full groups of ergodic p.m.p. equivalence relations, our result roughly says that such homomorphisms come only from actions of the equivalence relation, or of one of its symmetric powers. This has several rigidity consequences for homomorphisms between full groups. Our main application is a characterization of property (T) for ergodic p.m.p. equivalence relations purely in full group terms, without using their topology: an ergodic p.m.p. equivalence relation has (T) iff all non-free ergodic boolean actions of its full group are strongly ergodic.

math.GR↗

arXiv2026

Spaces of UCP maps and subalgebras of von Neumann algebras

In this paper, we establish that several natural topologies on the space of state-preserving unital completely positive maps coincide and that make it a Polish space. We then focus on the subspace of state-preserving conditional expectations and analyse its topology in detail, recovering the Haagerup-Winslow result that it aligns with the Effros-Maréchal topology on the space of von Neumann subalgebras. This correspondence is then applied to structural classes of subalgebras, including amenable, Haagerup and weakly amenable subalgebras. Among other consequences, we demonstrate the closedness of amenable subalgebras admitting state-preserving conditional expectations and analyze the semicontinuity and failure of continuity of the Cowling-Haagerup constant as a function on subalgebras. Finally, we investigate the space of von Neumann subalgebras that are not the image of state preserving conditional expectations for a fixed faithful normal state. For several important classes of von Neumann algebras, such as type ${\rm III}_λ$ factors with $0 \le λ< 1$ and type ${\rm III}_1$ factors with a state whose centraliser is infinite dimensional, we show that the subalgebras lacking state-preserving conditional expectations form an open and dense subset. Thus, in these settings, the generic subalgebra is not the range of state preserving conditional expectation.

math.OA↗

arXiv2026

On the Rokhlin lemma for infinite measure-preserving bijections

We study the Rokhlin lemma in the context of infinite measure-preserving bijections, and completely classify such bijections up to $λ$-approximate conjugacy, where $λ$ is the infinite measure which is preserved. This sharpens the classical version of the Rokhlin lemma, which only provides such a classification up to $μ$-approximate conjugacy where $μ$ is a probability measure equivalent to $λ$.

math.DS↗

arXiv2025

On the space of subgroups of Baumslag-Solitar groups II: High transitivity

We continue our study of the perfect kernel of the space of transitive actions of Baumslag-Solitar groups by investigating high transitivity. We show that actions of finite phenotype are never highly transitive, except when the phenotype is 1, in which case high transitivity is actually generic. In infinite phenotype, high transitivity is generic, except when |m|=|n| where it is empty. We also reinforce the dynamical properties of the action by conjugation on the perfect kernel that we had established in our first paper, replacing topological transitivity by high topological transitivity.

math.GR↗

arXiv2025

On dense orbits in the space of subequivalence relations

We first explain how to endow the space of subequivalence relations of any non-singular countable equivalence relation with a Polish topology, extending the framework of Kechris' recent monograph on subequivalence relations of probability measure-preserving (p.m.p.) countable equivalence relations. We then restrict to p.m.p. equivalence relations and discuss dense orbits therein for the natural action of the full group and of the automorphism group of the relation. Our main result is a characterization of the subequivalence relations having a dense orbit in the space of subequivalence relations of the ergodic hyperfinite p.m.p. equivalence relation. We also show that in this setup, all full groups orbits are meager. We finally provide a few Borel complexity calculations of natural subsets in spaces of subequivalence relations using a natural metric we call the uniform metric. This answers some questions from an earlier version of Kechris' monograph.

math.DS↗

arXiv2025

On the stability of hyperbolicity under quantitative measure equivalence

A well-known result of Shalom says that lattices in SO$(n,1)$ are $\mathrm{L}^p$ measure equivalent for all $p<n-1$. His proof actually yields the following stronger statement: the natural coupling resulting from a suitable choice of fundamental domains from a uniform lattice to a non-uniform one is $(\mathrm{L}^p,\mathrm{L}^{\infty})$. Moreover, it is easy to see that the coupling is cobounded: the fundamental domain of the uniform lattice is contained in a union of finitely many translates of the fundamental domain of the non-uniform one. The purpose of this note is to prove that this statement is sharp in the following sense: if a ME-coupling from a hyperbolic group to a non-hyperbolic group is cobounded and $(\mathrm{L}^p,\mathrm{L}^{\infty})$, then $p$ must be less than some $p_0$ only depending on the hyperbolic group.

math.GR↗

arXiv2025

$\mathrm{L}^1$ full groups of flows

We introduce the concept of an $\mathrm{L}^{1}$ full group associated with a measure-preserving action of a Polish normed group on a standard probability space. These groups carry a natural Polish group topology induced by an $\mathrm{L}^1$ norm. Our construction generalizes $\mathrm{L}^{1}$ full groups of actions of discrete groups, which have been studied recently by the first author. We show that under minor assumptions on the actions, topological derived subgroups of $\mathrm{L}^{1}$ full groups are topologically simple and -- when the acting group is locally compact and amenable -- are whirly amenable and generically two-generated. $\mathrm{L}^{1}$ full groups of actions of compactly generated locally compact Polish groups are shown to remember the $\mathrm{L}^{1}$ orbit equivalence class of the action. For measure-preserving actions of the real line (also often called measure-preserving flows), the topological derived subgroup of an $\mathrm{L}^{1}$ full groups is shown to coincide with the kernel of the index map, which implies that $\mathrm{L}^{1}$ full groups of free measure-preserving flows are topologically finitely generated if and only if the flow admits finitely many ergodic components. We also prove a reconstruction-type result: the $\mathrm{L}^{1}$ full group completely characterizes the associated ergodic flow up to flip Kakutani equivalence. Finally, we study the coarse geometry of the $\mathrm{L}^{1}$ full groups. The $\mathrm{L}^{1}$ norm on the derived subgroup of the $\mathrm{L}^{1}$ full group of an aperiodic action of a locally compact amenable group is proved to be maximal in the sense of C. Rosendal. For measure-preserving flows, this holds for the $\mathrm{L}^{1}$ norm on all of the $\mathrm{L}^{1}$ full group.

math.DS↗

arXiv2025

Spatial models for boolean actions in the infinite measure-preserving setup

We show that up to a null set, every infinite measure-preserving action of a locally compact Polish group can be turned into a continuous measure-preserving action on a locally compact Polish space where the underlying measure is Radon. We also investigate the distinction between spatial and boolean actions in the infinite measure-preserving setup. In particular, we extend Kwiatkowska and Solecki's Point Realization Theorem to the infinite measure setup. We finally obtain a streamlined proof of a recent result of Avraham-Re'em and Roy: Lévy groups cannot admit nontrivial continuous measure-preserving actions on Polish spaces when the measure is locally finite.

math.DS↗

arXiv2024

On the space of subgroups of Baumslag-Solitar groups I: perfect kernel and phenotype

Given a Baumslag-Solitar group, we study its space of subgroups from a topological and dynamical perspective. We first determine its perfect kernel (the largest closed subset without isolated points). We then bring to light a natural partition of the space of subgroups into one closed subset and countably many open subsets that are invariant under the action by conjugation. One of our main results is that the restriction of the action to each piece is topologically transitive. This partition is described by an arithmetically defined function, that we call the phenotype, with values in the positive integers or infinity. We eventually study the closure of each open piece and also the closure of their union. We moreover identify in each phenotype a (the) maximal compact invariant subspace.

math.GR↗

arXiv2024

Michael's selection theorem and applications to the Maréchal topology

The Maréchal topology, also called the Effros-Maréchal topology, is a natural topology one can put on the space of all von Neumann subalgebras of a given von Neumann algebra. It is a result of Maréchal from 1973 that this topology is Polish as soon as the ambient algebra has separable predual, but the sketch of proof in her research announcement appears to have a small gap. Our main goal in this paper is to fill this gap by a careful look at the topologies one can put on the space of weak-$*$ closed subspaces of a dual space. We also indicate how Michael's selection theorem can be used as a step towards Maréchal's theorem, and how it simplifies the proof of an important selection result of Haagerup and Winsløw for the Maréchal topology. As an application, we show that the space of finite von Neumann algebras is $\mathbfΠ^0_3$-complete.

math.OA↗

arXiv2023

Belinskaya's theorem is optimal

Belinskaya's theorem states that given an ergodic measure-preserving transformation, any other transformation with the same orbits and an $\mathrm{L}^1$ 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 $\mathrm{L}^p$. 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.

math.DS↗

arXiv2022

A characterization of high transitivity for groups acting on trees

We establish a sharp sufficient condition for groups acting on trees to be highly transitive when the action on the tree is minimal of general type. This gives new examples of highly transitive groups, including icc non-solvable Baumslag-Solitar groups, thus answering a question of Hull and Osin.

math.GR↗

arXiv2022

Quantitative measure equivalence between amenable groups

We initiate a quantitative study of measure equivalence (and orbit equivalence) between finitely generated groups, which extends the classical setting of $\mathrm L^p$ measure equivalence. In this paper, our main focus will be on amenable groups, for which we prove both rigidity and flexibility results. On the rigidity side, we prove a general monotonicity property satisfied by the isoperimetric profile, which implies in particular its invariance under $\mathrm L^1$ measure equivalence. This yields explicit "lower bounds" on how integrable a measure coupling between two amenable groups can be. This result also has an unexpected application to geometric group theory: the isoperimetric profile turns out to be monotonous under coarse embedding between amenable groups. This has various applications, among which the existence of an uncountable family of $3$-solvable groups which pairwise do not coarsely embed into one another. On the flexibility side, we construct explicit orbit equivalences between amenable groups with prescribed integrability conditions. Our main tool is a new notion of Følner tiling sequences. We show in a number of instances that the bounds derived from the isoperimetric profile are sharp up to a logarithmic factor. We also deduce from this study that two important quasi-isometry invariants are not preserved under $\mathrm L^1$ orbit equivalence: the asymptotic dimension and finite presentability.

math.GR↗

arXiv2022

Polish topologies on groups of non-singular transformations

In this paper, we prove several results concerning Polish group topologies on groups of non-singular transformation. We first prove that the group of measure-preserving transformations of the real line whose support has finite measure carries no Polish group topology. We then characterize the Borel $σ$-finite measures $λ$ on a standard Borel space for which the group of $λ$-preserving transformations has the automatic continuity property. We finally show that the natural Polish topology on the group of all non-singular transformations is actually its only Polish group topology.

math.GR↗

arXiv2021

On dense totipotent free subgroups in full groups

We study probability measure preserving (p.m.p.) non-free actions of free groups and the associated IRS's. The perfect kernel of a countable group Gamma is the largest closed subspace of the space of subgroups of Gamma without isolated points. We introduce the class of totipotent ergodic p.m.p. actions of Gamma: those for which almost every point-stabilizer has dense conjugacy class in the perfect kernel. Equivalently, the support of the associated IRS is as large as possible, namely it is equal to the whole perfect kernel. We prove that every ergodic p.m.p. equivalence relation R of cost $<r$ can be realized by the orbits of an action of the free group F_r on r generators that is totipotent and such that the image in the full group [R] is dense. We explain why these actions have no minimal models.This also provides a continuum of pairwise orbit inequivalent invariant random subgroups of F_r, all of whose supports are equal to the whole space of infinite index subgroups. We are led to introduce a property of topologically generating pairs for full groups (we call evanescence) and establish a genericity result about their existence. We show that their existence characterizes cost 1.

math.GR↗

arXiv2021

On a measurable analogue of small topological full groups II

We pursue the study of $\mathrm L^1$ full groups of graphings and of the closures of their derived groups, which we call derived $\mathrm L^1$ full groups. Our main result shows that aperiodic probability measure-preserving actions of finitely generated groups have finite Rokhlin entropy if and only if their derived $\mathrm L^1$ full group has finite topological rank. We further show that a graphing is amenable if and only if its $\mathrm L^1$ full group is, and explain why various examples of (derived) $\mathrm L^1$ full groups fit very well into Rosendal's geometric framework for Polish groups. As an application, we obtain that every abstract group isomorphism between $\mathrm L^1$ full groups of amenable ergodic graphings must be a quasi-isometry for their respective $\mathrm L^1$ metrics. We finally show that $\mathrm L^1$ full groups of rank one transformations have topological rank 2.

math.GR↗

arXiv2021

Property (T) for Roelcke precompact Polish groups (after Ibarlucía, building on work of Ben Yaacov and Tsankov)

We present a recent result of Ibarlucía stating that every Roelcke precompact Polish group has Kazhdan's property (T). This striking theorem builds on the characterization of Roelcke precompact Polish groups as automorphism groups of $\aleph_0$-categorical metric structures due to Ben Yaacov and Tsankov. We introduce the underlying concepts coming from model theory, and we outline Ibarlucía's proof in the (already new!) particular case of the group of measure-preserving transformations of a standard probability space.

math.GR↗

arXiv2021

Homogeneous actions on Urysohn spaces

We show that many countable groups acting on trees, including free products of infinite countable groups and surface groups, are isomorphic to dense subgroups of isometry groups of bounded Urysohn spaces. This extends previous results of the first and last author with Y. Stalder on dense subgroups of the automorphism group of the random graph. In the unbounded case, we also show that every free product of infinite countable groups arises as a dense subgroup of the isometry group of the rational Urysohn space.

math.GR↗