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- Title
Actions of tame abelian product groups.
- Authors
Allison, Shaun; Shani, Assaf
- 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 [Equivalence relations induced by actions of Polish groups, Trans. Amer. Math. Soc. 347 (1995) 4765–4777] gave a characterization for when G is tame. In [L. Ding and S. Gao, Non-archimedean abelian Polish groups and their actions, Adv. Math. 307 (2017) 312–343], Ding and Gao showed that for such G, the orbit equivalence relation must in fact be potentially Π 6 0 , while conjecturing that the optimal bound could be Π 3 0 . We show that the optimal bound is D ( Π 5 0) by constructing an action of such a group G which is not potentially Π 5 0 , and show how to modify the analysis of [L. Ding and S. Gao, Non-archimedean abelian Polish groups and their actions, Adv. Math. 307 (2017) 312–343] to get this slightly better upper bound. It follows, using the results of Hjorth et al. [Borel equivalence relations induced by actions of the symmetric group, Ann. Pure Appl. Logic 92 (1998) 63–112], 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.
- Subjects
ABELIAN groups; SET theory; MODEL theory; ORBITS (Astronomy); EQUIVALENCE relations (Set theory); MATHEMATICS
- Publication
Journal of Mathematical Logic, 2023, Vol 23, Issue 3, p1
- ISSN
0219-0613
- Publication type
Article
- DOI
10.1142/S0219061322500283