Abelian pro-countable groups and orbit equivalence relations
Abelian pro-countable groups and orbit equivalence relations
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阿贝尔可数群和轨道等价关系
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发表时间：
2014
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通讯作者：
Maciej Malicki
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作者：
Maciej Malicki
我们研究了可以定义为波兰亲可数群，具有不变度量的非阿基米德群或拟可数群的群，即具有积拓扑的可数离散群的闭子直积。结果表明，对于每一个非局部紧致的阿贝尔拟可数群G，存在G的一个闭子群L，以及G/L的一个闭非局部紧致的子群K，它是离散可数群的直接积。作为一个应用，我们证明了对于H/L形式的每一个阿贝尔波兰群G，其中H，L是Iso(X)的闭子群，X是一个局部紧致可分度量空间（例如G是阿贝尔的，拟可数的），如果G在波兰空间Y上的每一个连续作用都诱导出一个可约为可数类的轨道等价关系，那么G是局部紧致的。
We study groups that can be defined as Polish, pro-countable groups, as non-archimedean groups with an invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable, discrete groups, endowed with the product topology. We show, among other results, that for every non-locally compact, abelian quasi-countable group G there exists a closed subgroup L of G, and a closed, non-locally compact subgroup K of G/L which is a direct product of discrete, countable groups. As an application we prove that for every abelian Polish group G of the form H/L, where H,L are closed subgroups of Iso(X) and X is a locally compact separable metric space (e.g., G is abelian, quasi-countable), G is locally compact iff every continuous action of G on a Polish space Y induces an orbit equivalence relation that is reducible to an equivalence relation with countable classes.