Popa superrigidity and countable Borel equivalence relations
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.
countable universal equivalence relation
scientific article; zbMATH DE number 5544174
scientific article