It is a long-standing open question whether locally compact Polish
groups can be characterized as Polish groups all of whose continuous
actions on a Polish space give rise to an essentially countable orbit
equivalence relation. Here, an equivalence relation is called
essentially countable if it can be definably (i.e. in a Borel way)
reduced to a relation with countable classes - intuitively, it means
that it has relatively simple invariants. The implication from left to
right has been proved by A. Kechris in the early 90-ties. In this
talk, I will prove the converse for the class of 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 argument giving rise to a new criterion
for non-essential countability. If time permits, I will also discuss
how continuous logic enters the picture, leading to a new proof of
Kechris' result. The talk will be based on a joint paper with A.
Kechris, A. Panagiotopoulos, and J. Zielinski, and a joint paper with
A. Hallback and T. Tsankov.