Prime II$_1$ factors arising from actions of product groups

Abstract

We prove that any II$_1$ factor arising from a free ergodic probability measure preserving action $\Gamma\curvearrowright X$ of a product $\Gamma=\Gamma_1\times\dots\times\Gamma_n$ of icc hyperbolic, free product or wreath product groups is prime, provided $\Gamma_i\curvearrowright X$ is ergodic, for any $1\leq i\leq n.$ We also completely classify all the tensor product decompositions of a II$_1$ factor associated to a free ergodic probability measure preserving action of a product of icc, hyperbolic, property (T) groups. As a consequence, we derive a unique prime factorization result for such II$_1$ factors. Finally, we obtain a unique prime factorization theorem for a large class of II$_1$ factors which have property Gamma.