Unique Cartan decomposition for II_1 factors arising from arbitrary actions of free groups

TL;DR

This paper proves the uniqueness of Cartan subalgebras in II_1 factors from free group actions, establishing that such factors are distinguished by their group actions and are not isomorphic across different free groups.

Contribution

It establishes a unique Cartan decomposition for II_1 factors from free group actions, extending to a broad class of groups including free products of amenable groups.

Findings

Unique Cartan subalgebra for free group action factors

Non-isomorphism of factors from different free groups

Extension to free products of amenable groups

Abstract

We prove that for any free ergodic probability measure preserving action \F_n \actson (X,\mu) of a free group on n generators \F_n, 2 \leq n \leq \infty, the associated group measure space II_1 factor $L^\infty(X) \rtimes \F_n$ has L^\infty(X) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II_1 factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II_1 factors arising from arbitrary actions of a rather large family of groups, including all free products of amenable groups and their direct products.