Prime II1 factors arising from actions of product groups

Abstract

We prove that any II1 factor arising from a free ergodic probability measure preserving action Γ↷X of a product Γ=Γ1×…×Γn of icc hyperbolic, free product or wreath product groups is prime, provided Γi↷X is ergodic, for any 1≤i≤n. We also completely classify all the tensor product decompositions of a II1 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 II1 factors. Finally, we obtain a unique prime factorization theorem for a large class of II1 factors which have property Gamma.