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arXiv · 2305.08168

On the First-Order Free Group Factor Alternative

Isaac Goldbring·Jennifer Pi
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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.

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Isaac Goldbring, Jennifer Pi. 2023-05-14. On the First-Order Free Group Factor Alternative. https://arxiv.org/abs/2305.08168

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