2023/05/14 by Isaac Goldbring, Goldbring, Isaac, Jennifer Pi +1
Mathematics · #Advanced Operator Algebra Research #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2305.08168
We investigate the problem of elementary equivalence of the free group factors, that is, do all free group factors L(\mathbbFn) 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.