  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 $\forall \exists$-theories of
the interpolated free group factors are increasing, and use this to establish
that the dichotomy holds on the level of $\forall \exists$-theories. We
conclude with some observations on related problems.
