Actions of tame abelian product groups

A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation is Borel. When G = Pi(n)gamma(n) for countable abelian gamma(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, Nonarchimedean 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 Pi(0)(6), while conjecturing that the optimal bound could be Pi(0)(3). We show that the optimal bound is D(Pi(0)(5)) by constructing an action of such a group G which is not potentially Pi(0)(5), and show how to modify the analysis of [L. Ding and S. Gao, Nonarchimedean 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.

Automated summary

A Polish group G is considered tame if the corresponding orbit equivalence relation is Borel under any continuous action of G. Solecki provided a characterization for when G is tame, for the case when G = Pi(n)gamma(n) where gamma(n) is a countable abelian group. Ding and Gao showed that the orbit equivalence relation for such G must be potentially Pi(0)(6), conjecturing that the optimal bound could be Pi(0)(3).