Treeable equivalence relations
type: Item
instance of: scholarly article
zbMATH Open document ID: 1278.03081
DOI: 10.1142/S0219061312500031
author: Greg Hjorth
published in: Journal of Mathematical Logic
publication date: 2012-09-06T00:00:00Z
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 37A20
zbMATH DE Number: 6078207
zbMATH Keywords: Borel equivalence relation
zbMATH Keywords: Borel reducibility
zbMATH Keywords: countable equivalence relation
zbMATH Keywords: treeable equivalence relation
zbMATH Keywords: ergodic equivalence relation
zbMATH Keywords: orbit equivalence relation
zbMATH Keywords: free group
MaRDI profile type: Publication
cites work: Indecomposability of treed equivalence relations
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cites work: Amenable actions and almost invariant sets
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cites work: Q3687381
title: Treeable equivalence relations
review text: An equivalence relation \(E\) on a standard Borel space \(X\) is countable if all equivalence classes are countable. It is treeable if there is a Borel acyclic graph whose connected components are the \(E\)-classes. If \(X\) carries a standard Borel probability measure \(\mu\), \(E\) is measure-preserving if for each Borel bijection \(f\) between Borel sets \(A\) and \(B\) with \(f(x) E x\) for all \(x \in A\), we have \(\mu (A) = \mu (B)\). \(E\) is ergodic if any Borel \(E\)-invariant set is either null or conull. Given two equivalence relations \(E\) and \(F\) on standard Borel spaces \(X\) and \(Y\), \(E\) is Borel-reducible to \(F\) if there is a Borel function \(\theta : X \to Y\) such that \(x_1 E x_2\) iff \(\theta (x_1) F \theta (x_2)\) holds for all \(x_1, x_2 \in X\).NEWLINENEWLINEThe author shows that for \(n \geq 2\), there are standard Borel probability spaces \((X_s, \mu_s)_{s \in {\mathbb R}}\), each equipped with a free measure-preserving ergodic action \(a_s\) of the free group \({\mathbb F}_n\) on \(n\) generators, such that for any \(s \neq t\), the orbit equivalence relation \(E_{a_s}\) is not Borel-reducible to \(E_{a_t}\). In particular, there are continuum many treeable countable Borel equivalence relations which are incomparable with respect to Borel reducibility. This should be contrasted to the well-known result that there are only countably many smooth countable Borel equivalence relations and one which is hyperfinite and non-smooth. The existence of continuum many incomparable countable Borel equivalence relations was established earlier by \textit{S. Adams} and \textit{A. S. Kechris} [J. Am. Math. Soc. 13, No. 4, 903--943 (2000; Zbl 0952.03057)] but their examples are not treeable.
description: scientific article; zbMATH DE number 6078207