Related papers: Borel actions of Polish groups

We prove that every open $\sigma$-locally Polish groupoid $G$ is Borel equivalent to the groupoid of models on the Urysohn sphere $\mathbb{U}$ of an $\mathcal{L}_{\omega_1\omega}$-sentence in continuous logic. In particular, the orbit…

The aim of this paper is to prove ergodic decomposition theorems for probability measures quasi-invariant under Borel actions of inductively compact groups (Theorem 1) as well as for sigma-finite invariant measures (Corollary 1). For…

We study topological groups that can be defined as Polish, pro-countable abelian groups, as non-archimedean abelian groups or as quasi-countable abelian groups, i.e., Polish subdirect products of countable, discrete groups, endowed with the…

Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $\mu$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group…

This article reviews a generous sampling of both classical and more recent results on the interplay between measurable and topological dynamics. In the first part we have surveyed the strong analogies between ergodic theory and topological…

It is known that the topology of a Polish group is uniquely determined by its Borel structure and group operations, but this does not give us a way to find the topology. In this article we expand on this theorem and give a criterion for a…

Following Davies, Elekes and Keleti, we study measured sets, i.e. Borel sets $B$ in $\mathbb{R}$ (or in a Polish group) for which there is a translation invariant Borel measure assigning positive and \sigma-finite measure to $B$. We…

We resolve the topological version of the Erd\H{o}s Similarity conjecture introduced previously by Gallagher, Lai and Weber. We show that a set is topologically universal on ${\mathbb R}$ if and only if it is of strong measure zero. As a…

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…

We show that, on any given finite Borel measure space with the ambient space being a Polish metric space, every Borel real-valued function is almost a bounded, uniformly continuous function in the sense that for every $\varepsilon > 0$…

Under a mild definability assumption, we characterize the family of Borel actions $\Gamma \curvearrowright X$ of tsi Polish groups on Polish spaces that can be decomposed into countably-many actions admitting complete Borel sets that are…

A variation of the Scott analysis of countable structures is applied to actions of non-Archimedean TSI Polish groups acting continuously on a Polish spaces. We give results on the potential Borel complexity spectrum of such groups, and…

A weakly continuous near-action of a Polish group $G$ on a standard Lebesgue measure space $(X,\mu)$ is whirly if for every $A\subseteq X$ of strictly positive measure and every neighbourhood $V$ of identity in $G$ the set $VA$ has full…

We consider the isometry group of the infinite dimensional separable hyperbolic space with its Polish topology. This topology is given by the pointwise convergence. For non-locally compact Polish groups, some striking phenomena like…

In this thesis, we study the existence of universal objets of two differents types in the theory of topological groups and theirs actions on compacts spaces. In the first part, we contribute to the problem of existence of test spaces for…

We define and study Poissonian actions of Polish groups as a framework to Poisson suspensions, characterize them spectrally, and provide a complete characterization of their ergodicity. We further construct 'spatial' Poissonian actions,…

We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If $k \geq 1$ and $A, B \subseteq \mathbb{R}^k$ are bounded Borel…

Let $(G,\cdot)$ be a Polish group. We say that a set $X \subset G$ is Haar null if there exists a universally measurable set $U \supset X$ and a Borel probability measure $\mu$ such that for every $g, h \in G$ we have $\mu(gUh)=0$. We call…

We study topological realizations of countable Borel equivalence relations, including realizations by continuous actions of countable groups, with additional desirable properties. Some examples include minimal realizations on any perfect…

We prove that the homeomorphism problem for connected compact metric spaces is Borel bireducible with a universal orbit equivalence relation induced by a Borel action of a Polish group.