Actions of $\mathbb {F}_\infty$ whose II$_1$ factors and orbit equivalence relations have prescribed fundamental group

Abstract:

We show that given any subgroup $\mathcal {F}$ of $\mathbb {R}_+$ which is either countable or belongs to a certain “large” class of uncountable subgroups, there exist continuously many free ergodic measure-preserving actions $\sigma _i$ of the free group with infinitely many generators $\mathbb {F}_\infty$ on probability measure spaces $(X_i,\mu _i)$ such that their associated group measure space II$_1$ factors $M_i=\operatorname {L}^\infty (X_i) \rtimes _{\sigma _i} \mathbb {F}_\infty$ and orbit equivalence relations $\mathcal {R}_i=\mathcal {R} (\mathbb {F}_\infty {\overset {}{\curvearrowright }} X_i)$ have fundamental group equal to $\mathcal {F}$ and with $M_i$ (respectively $\mathcal {R}_i$) stably non-isomorphic. Moreover, these actions can be taken so that $\mathcal {R}_i$ has no outer automorphisms and any automorphism of $M_i$ is unitarily conjugate to an automorphism that acts trivially on the subalgebra $\operatorname {L}^\infty (X_i)$ of $M_i$.