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Building on work of Popa, Ioana, and Epstein--T\"{o}rnquist, we show that,for every nonamenable countable discrete group $\Gamma$, the relations ofconjugacy, orbit equivalence, and von Neumann equivalence of free ergodic (orweak mixing) measure preserving actions of $\Gamma$ on the standard atomlessprobability space are not Borel, thus answering questions of Kechris. This isan optimal and definitive result, which establishes a neat dichotomy with theamenable case, since any two free ergodic actions of an amenable group on thestandard atomless probability space are orbit equivalent by classical resultsof Dye and Ornstein--Weiss. The statement about conjugacy solves thenonamenable case of Halmos' conjugacy problem in Ergodic Theory, originallyposed by Halmos in 1956 for ergodic transformations.  In order to obtain these results, we study ergodic (or weak mixing)class-bijective extensions of a given ergodic countable probability measurepreserving equivalence relation $R$. When $R$ is nonamenable, we show that therelations of isomorphism and von Neumann equivalence of extensions of $R$ arenot Borel. When $R$ is amenable, all the extensions of $R$ are again amenable,and hence isomorphic by classical results of Dye and Connes--Feldman--Weiss.This approach allows us to extend the results about group actions mentionedabove to the case of nonamenable locally compact unimodular groups, via thestudy of their cross-section equivalence relations.

The complexity of conjugacy, orbit equivalence, and von Neumann equivalence of actions of nonamenable groups