Non-locally compact Polish groups and non-essentially countable orbit equivalence relations

It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. In the talk, I will answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-Archimedean Polish groups, for which there is an alternative, game-theoretic proof giving rise to a new criterion for non-essential countability. I will also discuss the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable. This is joint work with A. Kechris, A. Panagiotopoulos, and J. Zielinski. The details of the Zoom meeting will be sent separately.