Abelian pro-countable groups and non-Borel orbit equivalence relations

TL;DR

This paper characterizes certain classes of Polish abelian pro-countable groups based on the Borel complexity of orbit equivalence relations they induce, distinguishing tame and relatively tame groups.

Contribution

It provides a characterization of tame and relatively tame abelian pro-countable groups in terms of the Borel nature of their orbit equivalence relations.

Findings

Tame groups induce only Borel orbit equivalence relations.

Relatively tame groups ensure Borel orbit relations for diagonal actions.

Characterization of these groups within the class of Polish pro-countable abelian groups.

Abstract

We study topological groups that can be defined as Polish, pro-countable abelian groups, as non-archimedean abelian groups or as quasi-countable abelian groups, i.e., Polish subdirect products of countable, discrete groups, endowed with the product topology. We characterize tame groups in this class, i.e., groups such that all orbit equivalence relations induced by their continuous actions on Polish spaces are Borel, and relatively tame groups $G$, i.e., groups such that every diagonal action $\alpha \times \beta$ of $G$ induces a Borel orbit equivalence relation, provided that the actions $\alpha$, $\beta$ of $G$ are continuous, and induce Borel orbit equivalence relations.