Popa superrigidity and countable Borel equivalence relations
type: Item
instance of: scholarly article
title: Popa superrigidity and countable Borel equivalence relations
zbMATH Open document ID: 1162.03029
author: Simon Thomas
published in: Annals of Pure and Applied Logic
publication date: 2009-04-16T00:00:00Z
review text: The author presents some applications of Popa's Cocycle Superrigidity Theorem to the theory of countable Borel equivalence relations. In particular, it is shown that the universal countable Borel equivalence relation \(E_{\infty}\) is not essentially free. Let \(X\) be a standard Borel space and let \(G\) be a countable group. A Borel action of \(G\) on \(X\) is free if \(g \cdot x \neq x\) for every \(1\neq g \in G\) and \(x \in X\). A countable Borel equivalence relation \(E\) on \(X\) is free if there exists a countable group \(G\) with a free Borel action on \(X\) such that \(E\) is the orbit equivalence relation induced by \(G\); while \(E\) is essentially free if there exists a free countable Borel equivalence relation \(F\) such that \(E\) and \(F\) are Borel equivalent. An infinite group \(G\) is quasi-finite if every proper subgroup of \(G\) is finite. A central result of the paper is that the universal countable equivalence relation \(E_{\infty}\) is not essentially free. The author also shows that there exist both uncountably many free and also uncountably many non-essentially free countable Borel equivalence relations up to Borel bireducibility. It is obtained that the isomorphism relation on the space of torsion-free abelian groups of finite rank is not countable universal, and no countable Borel equivalence relation induced by a Borel action of a quasi-finite group is universal. The author obtains numerous additional results and recalls plenty of open problems of the area.
reviewed by: Tamás Mátrai
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 37A15
Mathematics Subject Classification ID: 37A20
zbMATH DE Number: 5544174
zbMATH Keywords: Borel equivalence relation
zbMATH Keywords: superrigidity
zbMATH Keywords: countable universal equivalence relation
zbMATH Keywords: torsion-free abelian group
MaRDI profile type: Publication
full work available at URL: j.apal.2007.08.003
OpenAlex ID: W4210814776
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DOI: 10.1016/J.APAL.2007.08.003
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description: scientific article; zbMATH DE number 5544174