Equivalence relations induced by actions of Polish groups

Abstract

We give an algebraic characterization of those sequences ( H n ) ({H_n}) of countable abelian groups for which the equivalence relations induced by Borel (or, equivalently, continuous) actions of H 0 × H 1 × H 2 × ⋯ {H_0} \times {H_1} \times {H_2} \times \cdots are Borel. In particular, the equivalence relations induced by Borel actions of H ω {H^\omega } , H H countable abelian, are Borel iff H ≃ ⊕ p ( F p × Z ( p ∞ ) n p ) H \simeq { \oplus _p}({F_p} \times \mathbb {Z}{({p^\infty })^{{n_p}}}) , where F p {F_p} is a finite p p -group, Z ( p ∞ ) \mathbb {Z}({p^\infty }) is the quasicyclic p p -group, n p ∈ ω {n_p} \in \omega , and p 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.