Logic Seminar

A Borel equivalence relation that is induced by an action of a Polish group is essentially countable if it admits a countable complete Borel section. The notion of (σ-)lacunarity strengthens essential countability by requiring the complete section to be uniformly separated within each orbit. Kechris proved that every action of a locally compact Polish group is lacunary. More recently, B. Miller found a G_0-type dichotomy that characterizes σ-lacunarity for actions of cli Polish groups. I will show that the notion of σ-lacunarity and essential countability coincide for Borel equivalence relations that are induced by actions of Polish groups. I will discuss some consequences for actions of tsi non-archimedean Polish groups.