$\sigma $-lacunary actions of Polish groups

Abstract

We show that every essentially countable orbit equivalence relation induced by a continuous action of a Polish group on a Polish space is $ \sigma $-lacunary. In combination with Gao and Jackson [Invent. Math. 201 (2015), pp. 309-383] we obtain a straightforward proof of the result from Ding and Gao [Adv. Math. 307 (2017), pp. 312-343] that every essentially countable equivalence relation that is induced by an action of an abelian nonarchimedean Polish group is Borel reducible to $ \mathbb{E}_0$, i.e., it is essentially hyperfinite.