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

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.