Equivalence relations induced by actions of Polish groups

Abstract:

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 ${H_0} \times {H_1} \times {H_2} \times \cdots$ are Borel. In particular, the equivalence relations induced by Borel actions of ${H^\omega }$, $H$ countable abelian, are Borel iff $H \simeq { \oplus _p}({F_p} \times \mathbb {Z}{({p^\infty })^{{n_p}}})$, 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 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.