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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm987-1-2016,
author = {Maciej Malicki},
title = {Abelian pro-countable groups and orbit equivalence relations},
journal = {Fundamenta Mathematicae},
volume = {233},
year = {2016},
pages = {83-99},
zbl = {06545391},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm987-1-2016}
}
Maciej Malicki. Abelian pro-countable groups and orbit equivalence relations. Fundamenta Mathematicae, Tome 233 (2016) pp. 83-99. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm987-1-2016/