Universal minimal flows of extensions of and by compact groups

TL;DR

This paper characterizes the structure of universal minimal flows for certain topological groups, showing how they decompose when the group has specific compact normal subgroups, with applications to totally disconnected locally compact Polish groups.

Contribution

It provides a new description of universal minimal flows for groups with compact normal subgroups, extending previous results and identifying their phase spaces explicitly.

Findings

Phase space of M(G) is homeomorphic to a product involving K and M(G/K).

Under certain conditions, the action on M(G) can be explicitly described.

For specific Polish groups, the phase space is a finite set, Cantor set, or their product.

Abstract

Every topological group $G$ has up to isomorphism a unique minimal $G$-flow that maps onto every minimal $G$-flow, the universal minimal flow $M(G).$ We show that if $G$ has a compact normal subgroup $K$ that acts freely on $M(G)$ and there exists a uniformly continuous cross section $G/K\to G,$ then the phase space of $M(G)$ is homeomorphic to the product of the phase space of $M(G/K)$ with $K$. Moreover, if either the left and right uniformities on $G$ coincide or $G\cong G/K\ltimes K$, we also recover the action, in the latter case extending a result of Kechris and Soki\'c. As an application, we show that the phase space of $M(G)$ for any totally disconnected locally compact Polish group $G$ with a normal open compact subgroup is homeomorphic to a finite set, Cantor set $2^{\mathbb{N}}$, $M(\mathbb{Z})$, or $M(\mathbb{Z})\times 2^{\mathbb{N}}.$