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

Abstract

We study topological groups G G for which either the universal minimal G G -system M ( G ) M(G) or the universal irreducible affine G G -system I A ( G ) I\!A(G) is tame. We call such groups “intrinsically tame” and “convexly intrinsically tame”, respectively. These notions, which were introduced in [Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics, Springer, Cham, 2018, pp. 351–392], are generalized versions of extreme amenability and amenability, respectively. When M ( G ) M(G) , as a G G -system, admits a circular order we say that G G is intrinsically circularly ordered. This implies that G G is intrinsically tame. We show that given a circularly ordered set X ∘ X_\circ , any subgroup G ≤ A u t ( X ∘ ) G \leq {\mathrm {A}ut}\,(X_\circ ) whose action on X ∘ X_\circ is ultrahomogeneous, when equipped with the topology τ p \tau _p of pointwise convergence, is intrinsically circularly ordered. This result is a “circular” analog of Pestov’s result about the extreme amenability of ultrahomogeneous actions on linearly ordered sets by linear order preserving transformations. We also describe, for such groups G G , the dynamics of the system M ( G ) M(G) , show that it is extremely proximal (whence M ( G ) M(G) coincides with the universal strongly proximal G G -system), and deduce that the group G G must contain a non-abelian free group. In the case where X X is countable, the corresponding Polish group of circular automorphisms G = A u t ( X o ) G={\mathrm {A}ut}\,(X_o) admits a concrete description. Using the Kechris–Pestov–Todorcevic construction we show that M ( G ) = S p l i t ( T ; Q ∘ ) M(G)={\mathrm {Split}}(\mathbb {T};\mathbb {Q}_{\circ }) , a circularly ordered compact metric space (in fact, a Cantor set) obtained by splitting the rational points on the circle T \mathbb {T} . We show also that G = A u t ( Q ∘ ) G={\mathrm {A}ut}\,(\mathbb {Q}_{\circ }) is Roelcke precompact, satisfies Kazhdan’s property T T (using results of Evans–Tsankov), and has the automatic continuity property (using results of Rosendal–Solecki).