On the first-order free group factor elementary equivalence

We investigate the problem of elementary equivalence of the free group factors, that is, do all free group factors L(Fn) 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 ∀∃-theories of the interpolated free group factors are increasing, and use this to establish that the dichotomy holds on the level of ∀∃-theories. We conclude with some observations on related problems.