Minimal model-universal flows for locally compact Polish groups

TL;DR

This paper extends the existence of minimal, model-universal flows from countable to all locally compact Polish groups, enabling the recovery of all free actions on standard Lebesgue spaces through invariant measures.

Contribution

It generalizes Weiss's result by constructing minimal, model-universal flows for every locally compact Polish group, broadening the scope of previous work.

Findings

Existence of minimal, model-universal flows for all locally compact Polish groups.

Extension of Weiss's result from countable groups to all locally compact Polish groups.

Framework for recovering free actions via invariant measures on these flows.

Abstract

Let $G$ be a locally compact Polish group. A metrizable $G$-flow $Y$ is called model-universal if by considering the various invariant probability measures on $Y$, we can recover every free action of $G$ on a standard Lebesgue space up to isomorphism. Weiss has shown that for countable $G$, there exists a minimal, model-universal flow. In this paper, we extend this result to all locally compact Polish groups.