It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

中文翻译：

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关于承认非本质可数行为的波兰团体

这是一个长期悬而未决的问题，是否每个不是局部紧致的波兰群都承认一个标准 Borel 空间上的 Borel 作用，其相关的轨道等价关系本质上是不可数的。对于嵌入局部紧致度量空间的等距群的所有波兰群，我们积极地回答了这个问题。该类包含所有非阿基米德波兰群，我们为此提供了基于非必要可数性新标准的替代证明。最后，我们提供 Solecki 定理的以下变体：每个无限维 Banach 空间都有一个连续作用，其轨道等价关系是 Borel 但本质上不可数。