Non-archimedean abelian Polish groups and their actions
Non-archimedean abelian Polish groups and their actions
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非阿基米德阿贝尔波兰群及其行为
DOI：
10.1016/j.aim.2016.11.019
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发表时间：
2017
影响因子：
1.7
通讯作者：
Su Gao
中科院分区：
文献类型：
--
作者：
Longyun Ding;Su Gao
本文考虑轨道等价关系都是Borel的非阿基米德阿贝尔波兰群。这样的群体被称为驯服。证明了非阿基米德阿贝尔波兰群G是tame的当且仅当对任意素数p，不存在从G的闭子群到Z ω或（Z（p）<ω）ω的连续满同态.除了确定tame群的结构外，我们还考虑了这类群的作用，并研究了它们在Borel约化谱中轨道等价关系的复杂性.它表明，如果这样的轨道等价关系是本质可数的，那么它必须是本质超有限的。我们还发现一个上界的Borel约化层次的轨道等价关系的所有驯服的非阿基米德阿贝尔波兰群。
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 Z ω or (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.