$σ$-Lacunary actions of Polish groups

We show that every essentially countable orbit equivalence relation induced by a continuous action of a Polish group on a Polish space is $σ$-lacunary. In combination with [Invent. Math.201 (1), 309-383, 2015] we obtain a straightforward proof of the result from [Adv. Math.307, 312-343,2017] that every essentially countable equivalence relation that is induced by an action of abelian non-archimedean Polish group is essentially hyperfinite.

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