Actions of F_\infty whose II_1 factors and orbit equivalence relations have prescribed fundamental group

TL;DR

This paper constructs many free ergodic actions of free groups with prescribed fundamental groups for their associated II_1 factors and orbit equivalence relations, demonstrating a high level of control over their automorphism structures.

Contribution

It shows the existence of numerous actions with specified fundamental groups and trivial outer automorphisms, expanding understanding of the structure of II_1 factors and orbit relations.

Findings

Existence of many actions with prescribed fundamental groups.

Construction of actions with no outer automorphisms.

Demonstration of stable non-isomorphism among constructed factors.

Abstract

We show that given any subgroup F of R_+ which is either countable or belongs to a certain "large" class of uncountable subgroups, there exist continuously many free ergodic probability measure preserving actions \sigma_i of the free group with infinitely many generators such that their associated group measure space II_1 factors M_i and orbit equivalence relations R_i have fundamental group equal to F and with M_i (respectively R_i) stably non-isomorphic. Moreover, these actions can be taken so that R_i has no outer automorphisms and any automorphism of M_i is unitary conjugate to an automorphism that acts trivially on $L^\infty(X_i) \subset M_i$.