We prove that for a countable discrete group Γ containing a copy of the free group ${\mathbb{F}}_n$, for some $2\le n\le \infty$, 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 relation in the spaces of separably acting factors of type ${\mathrm{II}}_1$, ${\mathrm{II}}_{\infty }$ and ${\mathrm{III}}_{\lambda }$, $0\le \lambda \le 1$, are analytic and not Borel when these spaces are given the Effros Borel structure.

中文翻译：

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冯·诺依曼等价的Borel复杂度

我们证明对于包含自由组副本的可数离散组Γ ${\mathbb{F}}_{ñ}$， 对于一些 $2\le ñ\le \infty$作为正常子群，Γ遍历ae无概率测度保持作用的共轭，轨道等价关系和von Neumann等价关系是波兰的概率测度保持Γ作用空间中的解析非Borel等价关系。结果，我们得到在类型分别起作用的因子的空间中的同构关系${\mathrm{II}}_{\mathrm{1个}}$， ${\mathrm{II}}_{\infty }$ 和 ${\mathrm{三级}}_{\lambda }$， $0\le \lambda \le \mathrm{1个}$当给这些空间赋予Effros Borel结构时，是而不是Borel。