Actions of tame abelian product groups

2021/05/11 by Shaun Allison, Allison, Shaun, Assaf Shani +1
Computer Science · Mathematics · #03E15 #03E25 #03E75 #54H05 #Computability, Logic, AI Algorithms #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2105.05144

openalex publication_date 2021/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation is Borel. When G = ∏_n Γ_n for countable abelian Γ_n, Solecki (1995) gave a characterization for when G is tame. Ding and Gao (2017) showed that for such G, the orbit equivalence relation must in fact be potentially \mathbfΠ⁰₆, while conjecturing that the optimal bound could be \mathbfΠ⁰₃. We show that the optimal bound is D(\mathbfΠ⁰₅) by constructing an action of such a group G which is not potentially \mathbfΠ⁰₅, and show how to modify the analysis of Ding and Gao to get this slightly better upper bound. It follows, using the results of Hjorth, Kechris, and Louvaeu (1998), that this is the optimal bound for the potential complexity of actions of tame abelian product groups. Our lower-bound analysis involves forcing over models of set theory where choice fails for sequences of finite sets.

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