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

Eli Glasner, Michael Megrelishvili

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We study topological groups G for which either the universal minimal G-system M(G) or the universal irreducible affine G-system IA(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), 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 given a circularly ordered set X◦, any subgroup G ≤ Aut (X) whose action on X is ultrahomogeneous, when equipped with the topology τp 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, the dynamics of the system M(G), show that it is extremely proximal (whence M(G) coincides with the universal strongly proximal G-system), and deduce that the group G must contain a non-abelian free group. In the case where X is countable, the corresponding Polish group of circular automorphisms G = Aut (Xo) admits a concrete description. Using the Kechris-Pestov-Todorcevic construction we show that M(G) = Split(T;Q), a circularly ordered compact metric space (in fact, a Cantor set) obtained by splitting the rational points on the circle T. We show also that G = Aut (Q) is Roelcke precompact, satisfies Kazhdan’s property T (using results of Evans-Tsankov), and has the automatic continuity property (using results of Rosendal-Solecki).

Original languageEnglish
Title of host publicationTopology, Geometry, and Dynamics
EditorsAnatoly M. Vershik, Victor M. Buchstaber, Andrey V. Malyutin
PublisherAmerican Mathematical Society
Pages133-154
Number of pages22
ISBN (Print)9781470456641
DOIs
StatePublished - 2021
EventInternational Conference on Topology, Geometry, and Dynamics, 2019 - St. Petersburg, Russian Federation
Duration: 19 Aug 201923 Aug 2019

Publication series

NameContemporary Mathematics
Volume772
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627

Conference

ConferenceInternational Conference on Topology, Geometry, and Dynamics, 2019
Country/TerritoryRussian Federation
CitySt. Petersburg
Period19/08/1923/08/19

Keywords

  • Amenability
  • Automatic continuity
  • Circular order
  • Extremely amenable
  • Fraïssé class
  • Intrinsically tame
  • Kazhdan’s property T
  • Roelcke precompact
  • Thompson’s circular group
  • Ultrahomogeneous

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