## 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 τ_{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, 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 (X_{o}) 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 language | English |
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Title of host publication | Topology, Geometry, and Dynamics |

Editors | Anatoly M. Vershik, Victor M. Buchstaber, Andrey V. Malyutin |

Publisher | American Mathematical Society |

Pages | 133-154 |

Number of pages | 22 |

ISBN (Print) | 9781470456641 |

DOIs | |

State | Published - 2021 |

Event | International Conference on Topology, Geometry, and Dynamics, 2019 - St. Petersburg, Russian Federation Duration: 19 Aug 2019 → 23 Aug 2019 |

### Publication series

Name | Contemporary Mathematics |
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Volume | 772 |

ISSN (Print) | 0271-4132 |

ISSN (Electronic) | 1098-3627 |

### Conference

Conference | International Conference on Topology, Geometry, and Dynamics, 2019 |
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Country/Territory | Russian Federation |

City | St. Petersburg |

Period | 19/08/19 → 23/08/19 |

### Bibliographical note

Publisher Copyright:© 2021 by the American Mathematical Society.

## Keywords

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