Topological dynamics beyond Polish groups

TL;DR

This paper introduces CAP groups, a new class of topological groups extending the metrizability dividing line in topological dynamics beyond Polish groups, and computes their universal minimal flows.

Contribution

It defines CAP groups, provides multiple characterizations, and applies these results to compute universal minimal flows for certain homeomorphism groups.

Findings

CAP groups generalize the metrizability dividing line

Characterizations of CAP groups in various forms

Universal minimal flows computed for specific homeomorphism groups

Abstract

When $G$ is a Polish group, metrizability of the universal minimal flow has been shown to be a robust dividing line in the complexity of the topological dynamics of $G$. We introduce a class of groups, the CAP groups, which provides a neat generalization of this dividing line to all topological groups. We prove a number of characterizations of this class, having very different flavors, and use these to prove that the class of CAP groups enjoys a number of nice closure properties. As a concrete application, we compute the universal minimal flow of the homeomorphism groups of several scattered topological spaces, building on recent work of Gheysens.