Abelian pro-countable groups and orbit equivalence relations

TL;DR

This paper investigates the structure of abelian pro-countable groups and their orbit equivalence relations, revealing conditions under which these groups are locally compact based on their actions.

Contribution

It establishes a structural characterization of non-locally compact abelian quasi-countable groups and links group properties to the complexity of their orbit equivalence relations.

Findings

Existence of specific closed subgroups within non-locally compact abelian quasi-countable groups.

Characterization of local compactness via orbit equivalence relations induced by group actions.

Connection between group structure and the reducibility of orbit equivalence relations.

Abstract

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.