The Complexity of Conjugacy, Orbit Equivalence, and von Neumann Equivalence of Actions of Nonamenable Groups

Let Γ be a nonamenable countable, discrete group. Building on work of Popa, Ioana, and Moroz (formerly Epstein) –Törnquist, we show that the relations of conjugacy, orbit equivalence, and von Neumann equivalence of free, ergodic (or weak mixing) measure preserving actions of Γ on the standard atomless probability space are not Borel. This answers questions of Kechris, and is an optimal and definitive result which establishes a neat dichotomy with the amenable case. Indeed, classical results of Dye and Ornstein–Weiss imply that any two free, ergodic actions of an amenable group on the standard atomless probability space are orbit equivalent. The statement about conjugacy solves the nonamenable case of Halmos’ conjugacy problem in Ergodic Theory, originally posed in 1956 for ergodic transformations. In order to obtain these results, we study ergodic (or weakly mixing) class-bijective extensions of a given ergodic countable probability measure preserving equivalence relation R R. When R R is nonamenable, we show that the relations of isomorphism and von Neumann equivalence of extensions of R R are not Borel. On the other hand, if R R is amenable then all the extensions of R 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 mentioned above to the case of nonamenable locally compact unimodular groups, via the study of their cross-section equivalence relations.