## Abelian pro-countable groups and orbit equivalence relations

### Volume 233 / 2016

#### Abstract

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 $L \leq G$ and a closed, non-locally compact $K \leq G/L$ 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 $H,L \leq {\rm Iso }(X)$ 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.