On the First-Order Free Group Factor Alternative
Isaac Goldbring, Jennifer Pi
TL;DR
The paper investigates whether all free group factors $L(\mathbb F_n)$ share a single first-order theory. It proves a trichotomy for their common first-order fundamental group and outlines potential routes to a full dichotomy mirroring the free group factor isomorphism problem, conditional on ultraproduct behavior. It also shows that the $\forall\exists$-theories of interpolated free group factors are monotone in the parameter $r$, yielding a $\forall\exists$-theory dichotomy under the trichotomy, and develops preservation-under-free-products and existential-embedding techniques (via Sela and Popa) to compare theories across the family. The work further connects these model-theoretic questions to matrix ultraproducts and reduced group C*-algebras, highlighting deep links between operator algebras, logic, and free probability, and outlining several open problems and plausible conjectures for advancing the dichotomy.
Abstract
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.