Actions of non-compact and non-locally compact Polish groups
Actions of non-compact and non-locally compact Polish groups
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非紧凑型和非局部紧凑型波兰团体的行动
DOI：
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发表时间：
2000
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通讯作者：
Slawomir Solecki
中科院分区：
文献类型：
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作者：
Slawomir Solecki
摘要 我们证明每个非紧波兰群都承认波兰空间上具有非光滑轨道等价关系的连续作用。我们实际上构建了一个免费的这样的动作。因此，对于波兰群体来说，紧凑性相当于该群体的所有连续自由动作都是平滑的。这回答了凯克里斯的问题。我们还建立了与该群的局部紧性相关的结果，该群无法导出不可简化为可数 Borel 等价关系的轨道等价关系。推广Hjorth的结果，我们证明了每个非局部紧性，即无限维可分Banach空间对具有非Borel轨道等价关系的Polish空间具有连续作用，从而表明该性质表征了Banach空间之间的非局部紧性。
Abstract We show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a result of Hjorth, we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces.