# Circular orders, ultra-homogeneous order structures and their automorphism groups

## TL;DR

This paper explores the properties of intrinsically tame topological groups with circular orderings, demonstrating their structural characteristics, automorphism groups, and connections to ultrahomogeneous actions and compact spaces.

## Contribution

It introduces the concept of intrinsically circularly ordered groups, links them to ultrahomogeneous actions, and provides a concrete description of their universal minimal systems.

## Key findings

- Groups are intrinsically circularly ordered when their actions are ultrahomogeneous on circularly ordered sets.
- The universal minimal system $M(G)$ is a circularly ordered compact space formed by splitting rational points on the circle.
- Groups are Roelcke precompact, have Kazhdan's property T, and possess the automatic continuity property.

## Abstract

We study topological groups $G$ for which the universal minimal $G$-system $M(G)$, or the universal irreducible affine $G$-system $IA(G)$ are tame. We call such groups intrinsically tame and convexly intrinsically tame. These notions are generalized versions of extreme amenability and amenability, respectively. When $M(G)$, as a $G$-system, admits a circular order we say that $G$ is intrinsically circularly ordered. This implies that $G$ is intrinsically tame. We show that for every circularly ultrahomogeneous action $G \curvearrowright X$ on a circularly ordered set $X$ the topological group $G$, in its pointwise convergence topology, is intrinsically circularly ordered. This result is a "circular" analog of Pestov's result about the extremal amenability of ultrahomogeneous actions on linearly ordered sets by linear order preserving transformations. In the case where $X$ is countable, the corresponding Polish group of circular automorphisms $G$ admits a concrete description. Using the Kechris-Pestov-Todorcevic construction we show that $M(G)$ is a circularly ordered compact space obtained by splitting the rational points on the circle. We show also that $G$ is Roelcke precompact, satisfies Kazhdan's property $T$ (using results of Evans-Tsankov) and has the automatic continuity property (using results of Rosendal-Solecki).