Borel equivalence relations which are highly unfree. (English) Zbl 1162.03028

A countable Borel equivalence relation \(E\) on a standard Borel space is essentially free if it is Borel reducible to an equivalence relation arising from the free Borel action of a countable group. It is shown in the paper under review that there is an ergodic, measure-preserving, countable Borel equivalence relation \(E\) on a standard Borel probability space \((X,\mu)\) such that \(E|_{C}\) is not essentially free on any conull \(C\subset X\). This answers a question of S. Thomas [see “Popa superrigidity and countable Borel equivalence relations”, Ann. Pure Appl. Logic 158, No. 3, 175–189 (2009; Zbl 1162.03029)].


03E15 Descriptive set theory
28D15 General groups of measure-preserving transformations
37A15 General groups of measure-preserving transformations and dynamical systems


Zbl 1162.03029
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