2014/05/04 by Maciej Malicki, Malicki, Maciej · 2 citations
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #math.GR #math.LO
paper · pdf · doi:10.48550/arxiv.1405.0693
arxiv created 2015/04/15 · arxiv updated 2015/04/16
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.