Orbit full groups for locally compact groups

TL;DR

This paper investigates the topological properties of orbit full groups generated by ergodic actions of non-discrete unimodular locally compact Polish groups, establishing their topological rank and amenability characteristics.

Contribution

It proves that such orbit full groups have topological rank two and characterizes their extreme amenability in relation to the acting group's amenability.

Findings

Orbit full groups have topological rank two.

Full groups generated by dense subgroups are dense in the orbit full group.

Extreme amenability of the full group corresponds to the group's amenability.

Abstract

We show that the topological rank of an orbit full group generated by an ergodic, probability measure-preserving free action of a non-discrete unimodular locally compact Polish group is two. For this, we use the existence of a cross section and show that for a locally compact Polish group, the full group generated by any dense subgroup is dense in the orbit full group of the action of the group. We prove that the orbit full group of a free action of a locally compact Polish group is extremely amenable if and only if the acting group is amenable, using the fact that the full group generates the von Neumann algebra of the action.