The Borel complexity of von Neumann equivalence

Abstract

We prove that for a countable discrete group Γ containing a copy of the free group Fn, for some 2≤n≤∞, as a normal subgroup, the equivalence relations of conjugacy, orbit equivalence and von Neumann equivalence of the ergodic a.e. free probability measure preserving 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 relations in the spaces of separably acting factors of type II1, II∞ and IIIλ, 0≤λ≤1, are analytic and not Borel when these spaces are given the Effros Borel structure.