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2016 | 233 | 1 | 83-99
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Abelian pro-countable groups and orbit equivalence relations

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We study a class of abelian groups that can be defined as Polish pro-countable groups, as non-archimedean groups with a compatible two-sided invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable discrete groups, endowed with the product topology.
We show that for every non-locally compact, abelian quasi-countable group G there exists a closed L ≤ G and a closed, non-locally compact K ≤ 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 ≤ Iso(X) and X is a locally compact separable metric space (in particular, for every abelian, quasi-countable group G), the following holds: 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.
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  • Department of Mathematics and Mathematical Economics, Warsaw School of Economics, al. Niepodległości 162, 02-554 Warszawa, Poland
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm987-1-2016
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