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Extremely amenable automorphism groups of countable structures

Mahmood Etedadialiabadi, Su Gao, Feng Li, Ruiwen Li

Abstract

In this paper we address the question: How many pairwise non-isomorphic extremely amenable groups are there which are separable metrizable or even Polish? We show that there are continuum many such groups. In fact we construct continuum many pairwise non-isomorphic extremely amenable groups as automorphism groups of countable structures. We also consider this classification problem from the point of view of descriptive set theory by showing that the class of all extremely amenable closed subgroups of $S_\infty$ is Borel and their isomorphism relation is more complex than any isomorphism relation of countable structures in the Borel reducibility hierarchy.

Extremely amenable automorphism groups of countable structures

Abstract

In this paper we address the question: How many pairwise non-isomorphic extremely amenable groups are there which are separable metrizable or even Polish? We show that there are continuum many such groups. In fact we construct continuum many pairwise non-isomorphic extremely amenable groups as automorphism groups of countable structures. We also consider this classification problem from the point of view of descriptive set theory by showing that the class of all extremely amenable closed subgroups of is Borel and their isomorphism relation is more complex than any isomorphism relation of countable structures in the Borel reducibility hierarchy.

Paper Structure

This paper contains 18 sections, 37 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topological dynamics
  4. Fraïssé Theory
  5. The KPT correspondence
  6. Descriptive set theory of equivalence relations
  7. The Extreme Amenability of Automorphism Groups
  8. $\Delta$-metric spaces
  9. The extreme amenability of $\hbox{\rm Aut}(\mathbb{U}_\Delta^<)$
  10. The extreme amenability of $\hbox{\rm Iso}(\mathbb{U}_\Delta)$
  11. Isomorphism Types of Extremely Amenable Groups
  12. Isomorphism types of $\hbox{\rm Iso}(\mathbb{U}_\Delta)$
  13. Isomorphism types of $\hbox{\rm Aut}(\mathbb{U}^<_\Delta)$
  14. The Complexity of the Equivalence between Distance Value Sets
  15. The exact complexity of $\approx_{\mathsf{CDVS}_+}$ and $\approx_{\mathsf{CDVS}}$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\Delta$ be a countable distance value set. Then the following hold:

Figures (1)

  • Figure 1: The composition of $D_1'$.

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1: Hodges
  • Theorem 2.1: Kechris-Pestov-Todorcevic KPT
  • ...and 57 more