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

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 all of whose continuous actions on a Polish space induce a Borel orbit equivalence relation, and relatively tame groups, i.e., groups all of whose diagonal actions induce a Borel orbit equivalence relation, provided that are continuous actions inducing Borel orbit equivalence relations.