Related papers: Polish group actions and effectivity

We present a comprehensive theory of boundedness properties for Polish groups developed with a main focus on Roelcke precompactness (precompactness of the lower uniformity) and Property (OB) (boundedness of all isometric actions on…

We study Polish spaces for which a set of possible distances $A \subseteq \mathbb{R}^+$ is fixed in advance. We determine, depending on the properties of $A$, the complexity of the collection of all Polish metric spaces with distances in…

We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher…

Let G be a closed subgroup of the group of all permutations of a countably infinite set. Let X be a Polish G-space with a countable basis A of clopen sets. Each x from X defines a characteristic function f on A by f(U)=1 iff x belongs to U…

We consider the notion of Borel reducibility between pseudometrics on standard Borel spaces introduced and studied recently by C\'{u}th, Doucha and Kurka, as well as the notion of an orbit pseudometric, a continuous version of the notion of…

The aim of this paper is to classify cohomogeneity one isometric actions on the 4-dimensional Minkowski space $\mathbb{R}^{3,1}$, up to orbit equivalence. Representations, up to conjugacy, of the acting groups in $O(3,1)\ltimes…

We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence…

Given an ergodic probability measure preserving dynamical system $\G\acts (X,\mu)$, where $\G$ is a finitely generated countable group, we show that the asymptotic growth of the number of finite models for the dynamics, in the sense of…

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…

It is shown that, in the Gromov space of isometry classes of pointed proper metric spaces, the equivalence relations defined by existence of coarse quasi-isometries or being at finite Gromov-Hausdorff distance, cannot be reduced to the…

We consider various notions of equivalence in the space of bounded operators on a Hilbert space, in particular modulo finite rank, modulo Schatten $p$-class, and modulo compact. Using Hjorth's theory of turbulence, the latter two are shown…

We prove that if the universal minimal flow of a Polish group $G$ is metrizable and contains a $G_\delta$ orbit $G \cdot x_0$, then it is isomorphic to the completion of the homogeneous space $G/G_{x_0}$ and show how this result translates…

We show that orbit equivalence relations arising from essentially free ergodic probability measure preserving actions of Zariski dense discrete subgroups of simple algebraic groups are strongly prime. As a consequence, we prove the…

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…

Working in the framework of Borel reducibility, we study various notions of embeddability between groups. We prove that the embeddability between countable groups, the topological embeddability between (discrete) Polish groups, and the…

We prove that orbit equivalence relations (ERs, for brevity) of generically turbulent Polish actions are not Borel reducible to ER s of a family which includes Polish actions of S_\infty, the group of all permutations of N, and is closed…

We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither…

We obtain a partial converse of Vershik's description of ergodic probability measures on a compact metric space with respect to an isometric action by an inductively compact group. This allows us to identify, in this setting, the set of…

We characterise the groupoid $C^*$-algebras associated to the transformation groupoids of injective actions of discrete countable Ore semi-groups on compact topological spaces in terms of the reduced crossed product from the dual actions,…

We prove that any divisible dynamical simplex is the set of invariant measures of some Toeplitz subshift. We apply our construction to prove that orbit equivalence of Toeplitz subshifts is Borel bireducible to the universal equivalence…