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

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