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 $\Gamma ={\Gamma }_1\times \cdots \times {\Gamma }_n$ of icc hyperbolic, free product or wreath product groups is prime, provided ${\Gamma }_i\curvearrowright X$ is ergodic, for any $1\le i\le n.$ 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.

To appear in Journal of Functional Analysis