Journal of Operator Theory

Volume 94, Issue 1, Summer 2025 pp. 129-150.

On the first-order free group factor elementary equivalence

Authors: Isaac Goldbring (1), Jennifer Pi (2)
Author institution: (1) Department of Mathematics, University of California - Irvine, Irvine, 92697, U.S.A.
(2) Department of Mathematics, University of California - Irvine, Irvine, 92697, U.S.A

Summary: 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.

DOI: http://dx.doi.org/10.7900/jot.2023sep11.246
Keywords: free group factors, noncommutative Tarski problem, existential embedding, fundamental group

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