Actions of tame abelian product groups

A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation Borel. When [Formula: see text] countable abelian text], Solecki [Equivalence relations induced by actions groups, Trans. Amer. Math. Soc. 347 (1995) 4765–4777] gave a characterization when tame. In [L. Ding and S. Gao, Non-archimedean groups their actions, Adv. 307 (2017) 312–343], Gao showed that such must in fact be potentially while conjecturing optimal bound could text]. We show constructing an which not how to modify analysis 312–343] get this slightly better upper bound. It follows, using results Hjorth et al. [Borel symmetric group, Ann. Pure Appl. Logic 92 (1998) 63–112], potential complexity product groups. Our lower-bound involves forcing over models set theory where choice fails sequences finite sets.