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

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.