On Polish groups admitting non-essentially countable actions

TL;DR

This paper investigates conditions under which Polish groups admit non-essentially countable Borel actions, providing positive results for certain classes and a new criterion, with implications for Banach space actions.

Contribution

It proves that all Polish groups embedding into isometry groups of locally compact spaces admit such actions and offers a new criterion for non-essential countability, extending to Banach spaces.

Findings

All Polish groups embedding into isometry groups of locally compact spaces admit non-essentially countable actions.

A new criterion for non-essential countability is introduced for non-archimedean Polish groups.

Every infinite-dimensional Banach space admits a Borel action with a non-essentially countable orbit equivalence relation.

Abstract

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. We 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 we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide 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.