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.

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).