Key subgroups in the Polish group of all automorphisms of the rational circle

TL;DR

This paper investigates the structure of certain subgroups within the Polish group of all automorphisms of the rational circle, revealing new properties about their topological and minimality characteristics.

Contribution

It identifies specific subgroups that are inj-key but not co-minimal, providing counterexamples to previous assumptions and addressing open questions in the field.

Findings

Certain subgroups are inj-key but not co-minimal in G

Counterexamples to previous conjectures about subgroup properties

Addresses open questions related to Polish groups and minimal flows

Abstract

Extending some results of a joint work with E. Glasner (2021) we continue to study the Polish group $G:=\mathrm{Aut}(\mathbb{Q}_0)$ of all circular order preserving permutations of $\mathbb{Q}_0$ with the pointwise topology, where $\mathbb{Q}_0$ is the rational discrete circle. We show that certain extremely amenable subgroups $H$ of $G:=\mathrm{Aut}(\mathbb{Q}_0)$ are inj-key (i.e., $H$ distinguishes weaker Hausdorff group topologies on $G$) but not co-minimal in $G$. This counterexample answers a question from a joint work with M. Shlossberg (2024) and is inspired by a question proposed by V. Pestov about Polish groups $G$ with metrizable universal minimal $G$-flow $M(G)$. It is an open problem to study Pestov's question in its full generality.