Equivalence relations induced by actions of polish groups

We give an algebraic characterization of those sequences of countable abelian groups for which the equivalence relations induced by Borel (or, equivalently, continuous) actions of are Borel. In particular, the equivalence relations induced by Borel actions of countable abelian, are Borel where Fp is a finite p-group, is the quasicyclic p-group, and p varies over the set of all primes. This answers a question of R. L. Sami by showing that there are Borel actions of Polish abelian groups inducing non-Borel equivalence relations. The theorem also shows that there exist non-locally compact abelian Polish groups all of whose Borel actions induce only Borel equivalence relations. In the process of proving the theorem we generalize a result of Makkai on the existence of group trees of arbitrary height.