{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:21:31Z","timestamp":1753881691648,"version":"3.41.2"},"reference-count":18,"publisher":"World Scientific Pub Co Pte Ltd","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:p> A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation is Borel. When [Formula: see text] for countable abelian [Formula: see text], Solecki [Equivalence relations induced by actions of Polish groups, Trans. Amer. Math. Soc.\u00a0347 (1995) 4765\u20134777] gave a characterization for when G is tame. In [L. Ding and S. Gao, Non-archimedean abelian Polish groups and their actions, Adv. Math.\u00a0307 (2017) 312\u2013343], Ding and Gao showed that for such G, the orbit equivalence relation must in fact be potentially [Formula: see text], while conjecturing that the optimal bound could be [Formula: see text]. We show that the optimal bound is [Formula: see text] by constructing an action of such a group G which is not potentially [Formula: see text], and show how to modify the analysis of [L.\u00a0Ding and S.\u00a0Gao, Non-archimedean abelian Polish groups and their actions, Adv. Math.\u00a0307 (2017) 312\u2013343] to get this slightly better upper bound. It follows, using the results of Hjorth et\u00a0al. [Borel equivalence relations induced by actions of the symmetric group, Ann. Pure Appl. Logic\u00a092 (1998) 63\u2013112], 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. <\/jats:p>","DOI":"10.1142\/s0219061322500283","type":"journal-article","created":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T08:25:23Z","timestamp":1667982323000},"source":"Crossref","is-referenced-by-count":0,"title":["Actions of tame abelian product groups"],"prefix":"10.1142","volume":"23","author":[{"given":"Shaun","family":"Allison","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA"}]},{"given":"Assaf","family":"Shani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Harvard University, Cambridge, MA 02138, USA"}]}],"member":"219","published-online":{"date-parts":[[2023,2,28]]},"reference":[{"key":"S0219061322500283BIB001","series-title":"London Mathematical Society Lecture Note Series","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511735264","volume-title":"The Descriptive Set Theory of Polish Group Actions","volume":"232","author":"Becker H.","year":"1996"},{"issue":"3","key":"S0219061322500283BIB002","doi-asserted-by":"crossref","first-page":"2250015","DOI":"10.1142\/S0219061322500155","volume":"22","author":"Clemens J. 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