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# Title: Circular orders, ultrahomogeneity and topological groups

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).
 Comments: 17 pages Subjects: Dynamical Systems (math.DS) MSC classes: Primary 37Bxx, Secondary 54H20, 54H15, 22A25 Cite as: arXiv:1803.06583 [math.DS] (or arXiv:1803.06583v1 [math.DS] for this version)

## Submission history

From: Michael Megrelishvili [view email]
[v1] Sat, 17 Mar 2018 22:53:48 GMT (23kb)