The Borel complexity of von Neumann equivalence

Abstract

We prove that for a countable discrete group $\Gamma$ containing a copy of the free group $\F_n$, for some $2\leq n\leq\infty$, as a normal subgroup, the equivalence relations of conjugacy, orbit equivalence and von Neumann equivalence of the ergodic a.e. free actions of $\Gamma$ are analytic non-Borel equivalence relations in the Polish space of probability measure preserving $\Gamma$ actions. As a consequence we obtain that the isomorphism relation in the spaces of separably acting factors of type $\II_1$, $\II_\infty$ and $\III_\lambda$, $0\leq\lambda\leq 1$, are analytic and not Borel when these spaces are given the Effros Borel structure.