  In this paper we consider non-archimedean abelian Polish groups whose orbit
equivalence relations are all Borel. Such groups are called tame. We show that
a non-archimedean abelian Polish group is tame if and only if it does not
involve $\Z^\omega$ or $(\Z(p)^{<\omega})^\omega$ for any prime $p$. In
addition to determining the structure of tame groups, we also consider the
actions of such groups and study the complexity of their orbit equivalence
relations in the Borel reducibility hierarchy. It is shown that if such an
orbit equivalence relation is essentially countable, then it must be
essentially hyperfinite. We also find an upper bound in the Borel reducibility
hierarchy for the orbit equivalence relations of all tame non-archimedean
abelian Polish groups.
