Non-archimedean abelian Polish groups and their actions

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• 作者：Longyun Ding^a ; ^{dinglongyun@gmail.com} ; Su Gao^b ; ^sgao@unt.edu
• 关键词：primary ; 03E15 ; 22A05 ; secondary ; 54H05 ; 54H11
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：307
• 期：Complete
• 页码：312-343
• 全文大小：561 K
• 卷排序：307

文摘

In this paper we consider non-archimedean abelian Polish groups whose orbit equivalence relations are all Borel. Such groups are called tame. We show that a non-archimedean abelian Polish group G is tame if and only if there does not exist a continuous surjective homomorphism from a closed subgroup of G   onto ZωZω or (Z(p)<ω)ω(Z(p)<ω)ω for any prime p. In addition to determining the structure of tame groups, we also consider the actions of such groups and study the complexity of their orbit equivalence relations in the Borel reducibility hierarchy. It is shown that if such an orbit equivalence relation is essentially countable, then it must be essentially hyperfinite. We also find an upper bound in the Borel reducibility hierarchy for the orbit equivalence relations of all tame non-archimedean abelian Polish groups.