Unique Cartan decomposition for II1 factors arising from arbitrary actions of free groups

Abstract

We prove that for any free ergodic probability measure-preserving action ${\mathbb{F}}_n\curvearrowright (X,\mathit{\mu })$ of a free group on n generators ${\mathbb{F}}_n,2\le n\le \infty$, the associated group measure space II₁ factor $L^{\infty }(X)⋊{\mathbb{F}}_n$ has L^∞(X) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II₁ factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II₁ factors arising from arbitrary actions of a much larger family of groups, including all free products of amenable groups and their direct products.