Summary: We give an algebraic characterization of those sequences \((H_n)\) of countable abelian groups for which the equivalence relations induced by Borel (or, equivalently, continuous) actions of \(H_0 \times H_1 \times H_2 \times \dots\) are Borel. In particular, the equivalence relations induced by Borel actions of \(H^\omega\), \(H\) countable abelian, are Borel iff \(H\simeq \bigoplus_p (F_p \times \mathbb{Z} (p^\infty)^{n_p})\), where \(F_p\) is a finite \(p\)-group, \(\mathbb{Z} (p^\infty)\) is the quasicyclic \(p\)-group, \(n_p\in \omega\), 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 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.