On the First-Order Free Group Factor Alternative

We investigate the problem of elementary equivalence of the free group factors, that is, do all free group factors $L({\mathbb{F}}_n)$ share a common first-order theory? We establish a trichotomy of possibilities for their common first-order fundamental group, as well as several possible avenues for establishing a dichotomy in direct analog to the free group factor alternative of Dykema and Radulescu. We also show that the $\mathrm{\forall }\mathrm{\exists }$-theories of the interpolated free group factors are increasing, and use this to establish that the dichotomy holds on the level of $\mathrm{\forall }\mathrm{\exists }$-theories. We conclude with some observations on related problems.