I will present the notion of hyper-u-amenability for countable Borel equivalence relations, a property that implies 1-amenability and which is automatic for orbit equivalence relations of continuous amenable actions on sigma-compact Polish spaces, and for orbit equivalence relations of Borel actions of amenable groups. I will then show that hyper-u-amenable, treeable countable Borel equivalence relations are hyperfinite. As corollaries, I will show that, for orbit equivalence relations of free continuous actions of free groups on sigma-compact spaces, measure-hyperfiniteness implies hyperfiniteness, and that the orbit equivalence relation of a Borel action by an amenable group is hyperfinite, if treeable. The material presented is part of a joint work with Petr Naryshkin.
We study measure-class-preserving (mcp) equivalence relations and seek criteria for their (non)amenability. Such criteria are well established for probability-measure-preserving (pmp) equivalence relations, where tools like cost and \(\ell^2\)-Betti numbers are available. However, in the mcp setting, these tools are absent and much less is known. We discuss a recently developed structure theory for mcp equivalence relations, including a precise characterization of amenability for treed mcp equivalence relations in terms of the interplay between the geometry of the trees and the Radon–Nikodym cocycle. This generalizes Adams’ dichotomy to the mcp setting and yields a complete description of the structure of amenable subrelations of treed equivalence relations, as well as anti-treeability results. We also establish a Day–von Neumann-type result for multi-ended mcp graphs, generalizing a theorem of Gaboriau and Ghys. Joint work with Robin Tucker-Drob, and with Ruiyuan Chen and Grigory Terlov.
Let Γ be a countably infinite discrete group. A Γ-flow X (i.e., a nonempty compact Hausdorff space equipped with a continuous action of Γ) is called S-minimal for a subset S⊆Γ if the partial orbit S⋅x is dense for every point x∈X. (When S=Γ, we recover the usual notion of minimality.) Despite the simplicity of the definition, given a group Γ, finding an S-minimal dynamical system is typically quite difficult (in particular even when Γ is the free group and S is a subgroup it was not previously known). In this talk, I will discuss a very recent result on how to construct S-minimal systems for any countable collection of infinite subsets simultaneously. Although the problem is purely dynamical, the techniques make heavy use of recent ideas from descriptive set theory. Indeed, once the main result is established, we can return to derive some non-obvious, purely Borel, corollaries. This is joint work with Anton Bernshteyn.
In the probability measure preserving (p.m.p.) setup, a well-known consequence of Rokhlin's lemma is that any two aperiodic p.m.p. bijections are approximately conjugate, meaning that up to conjugacy, they only differ on sets of arbitrarily small measure. In the infinite measure-preserving (i.m.p.) setup, we will see why this is false and characterize approximate conjugacy for aperiodic i.m.p. bijections. This is joint work with Fabien Hoareau.
A characterisation of isometry topological groups of Polish ultrametric spaces is provided. This answers in the context of Polish spaces a problem of Krasner's. A finer analysis allows also to characterise isometry topological groups of subclasses of Polish ultrametric spaces, providing a solution to questions raised by Gao and Kechris. The groups obtained involve various kinds of generalised wreath products proposed in the literature by Hall, Holland, and Malicki. This is a joint work with A. Marcone and L. Motto Ros.
Given a class of groups, say the class of all countably infinite groups, it is natural to ask: what group properties are generic in the sense of Baire category in this class? For this question to make sense, we need a topology. In the case of countably infinite groups, a natural choice is the following: fix \(\mathbb{N}\) as a common universe and identify every group on \(\mathbb{N}\) with the group operation that defines it. These group operations form a Polish subspace in \(\mathbb{N}^{\mathbb{N}\times\mathbb{N}}\). I will talk about generic group properties in this setting, and, if time permits, I will also say a few words about other settings and generic properties in various classes of topological groups.
We show that there exists an equidecomposition between a closed disk and a closed square of the same area in \(\mathbb{R}^2\) by translations with algebraic irrational coordinates. Our proof uses a new method for bounding the discrepancy of product sets in the \(k\)-torus using only the Erdős–Turán inequality. This resolves a question of Laczkovich from 1990. We also obtain an improved upper bound on the number of pieces required to square the circle, by proving effective bounds on such discrepancy estimates for translations by certain algebraic irrational numbers. This builds on an idea of Frank Calegari for bounding certain sums of products of fractional parts of algebraic numbers, and some computer assistance. This is joint work with Spencer Unger.