The Borel complexity of von Neumann equivalence

We prove that for a countable discrete group $Γ$ 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 $Γ$ are analytic non-Borel equivalence relations in the Polish space of probability measure preserving $Γ$ actions. As a consequence we obtain that the isomorphism relation in the spaces of separably acting factors of type $\II_1$, $\II_\infty$ and $\III_λ$, $0\leqλ\leq 1$, are analytic and not Borel when these spaces are given the Effros Borel structure.

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