Countable Borel Equivalence Relations

Abstract

This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces (i.e., Polish spaces equipped with their Borel structure). In mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants. Frequently these objects can be viewed as elements of a standard Borel space X and the equivalence turns out to be a Borel equivalence relation E on X. A complete classification of X up to E consists of finding a set of invariants I and a map c : X →

Théorème de Kurosh pour les relations d’équivalence boréliennes

“…De plus, quitte à renuméroter les fibres de F, on peut toujours supposer que dans chaque fibre la numérotation commence à 1 et ne saute pas d'entiers naturels ; • si f est une réduction de R à R , c'est-à-dire une application borélienne f : X −→ X telle que deux éléments de X sont R-équivalents si et seulement si leurs images par f sont R -équivalents (cf. [7]), alors il existe un domaine complet A de R tel que la restriction de f à A soit une équivalence orbitale entre…”

“…Corollaire 15 (voir aussi [7] et [4]). -Une sous-relation d'une relation d'équivalence borélienne arborable est arborable.…”

“…prop. 20, voir aussi [7] ou [4]), on en déduit que si R | A est arborable, alors il en est de même de R .…”

Borel equivalence relations and cardinal algebras

“…Consider the class T * of Borel equivalence relations which ares treeable and essentially countable. Then by Hjorth [H1] every such relation E admits a Borel countable complete section A and then by an argument similar to that in the proof of [JKL,Theorem 3.3 (i)] it follows that E|A is a treeable countable Borel equivalence relation and of course E|A ∼ B E. Therefore [T ] = [T * ] and part (iii) of Theorem 3.10 holds as well for T * .…”

Finite unions of equivalence relations

“…An equivalence relation E on X is hyperfinite if there is an increasing sequence F n n∈N of finite Borel subequivalence relations of E such that E = n∈N F n (equivalently, if there is a Borel action of Z on X that induces E). For more on countable Borel equivalence relations in general, see Jackson-Kechris-Louveau [2]. Despite being the object of much scrutiny in recent years, many basic questions about hyperfiniteness remain open.…”