Key subgroups in the Polish group of all automorphisms of the rational circle
Michael Megrelishvili
TL;DR
The paper investigates the Polish group $G = \operatorname{Aut}(\mathbb{Q}_0)$ of circular order preserving permutations with the pointwise topology, showing that extremely amenable subgroups $H$ can be inj-key without being co-minimal in $G$. It constructs concrete counterexamples, notably $H = G_{a_0}$ (often isomorphic to $\operatorname{Aut}(\mathbb{Q},\le)$), to demonstrate inj-key while failing co-minimality, addressing questions from MeSh-key and linking to Pestov's inquiry about metrizable universal minimal flows $M(G)$. The work leverages a dense embedding of $G$ into $\mathrm{Homeo}_+ (\mathbb{T})$, analyzes the associated coset-topologies, and identifies the four-orbit structure of $M(G)$ (with orbits $\mathbb{Q}_0^-$, $\mathbb{Q}_0^+$, and $J$). It also discusses how these findings relate to Pestov's broader question and to the structure of the completion $\beta_G(\mathbb{Q}_0)$, suggesting that inj-key does not imply co-minimal even in geometrically natural Polish groups.
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.