Metrizable universal minimal flows of Polish groups have a comeagre orbit

TL;DR

This paper proves that for Polish groups with metrizable universal minimal flows, there always exists a comeagre orbit, implying a specific structure for the flow involving an extremely amenable subgroup.

Contribution

It establishes the existence of a comeagre orbit in the universal minimal flow of certain Polish groups and characterizes the flow as a completion of a quotient by an extremely amenable subgroup.

Findings

Existence of a comeagre orbit in the universal minimal flow for certain Polish groups.

Universal minimal flow is isomorphic to a completion of a quotient by an extremely amenable subgroup.

Provides structural insight into the dynamics of Polish groups with metrizable flows.

Abstract

We prove that, whenever $G$ is a Polish group with metrizable universal minimal flow $M(G)$, there exists a comeagre orbit in $M(G)$. It then follows that there exists an extremely amenable, closed, coprecompact $G^*$ of $G$ such that $M(G) = \hat{G/G^*}$.