More Polish full groups

TL;DR

This paper introduces the concept of orbit full groups associated with Polish group actions, explores their properties, and demonstrates their role as invariants of orbit equivalence, with new examples and characterizations.

Contribution

It defines orbit full groups for Polish group actions, extends known concepts to new settings, and characterizes their topological and representation-theoretic properties.

Findings

Orbit full groups are complete invariants of orbit equivalence.

Ergodic orbit full groups have a unique Polish group topology.

Characterization of ergodic full groups via non-trivial continuous characters.

Abstract

We associate to every action of a Polish group on a standard probability space a Polish group that we call the orbit full group. For discrete groups, we recover the well-known full groups of pmp equivalence relations equipped with the uniform topology. However, there are many new examples, such as orbit full groups associated to measure preserving actions of locally compact groups. In fact, we show that such full groups are complete invariants of orbit equivalence. We give various characterizations of the existence of a dense conjugacy class for orbit full groups, and we show that the ergodic ones actually have a unique Polish group topology. Furthermore, we characterize ergodic full groups of countable pmp equivalence relations as those admitting non-trivial continuous character representations.