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Group topologies on automorphism groups of homogeneous structures

Zaniar Ghadernezhad, Javier de la Nuez González

Abstract

We classify all group topologies coarser than the topology of stabilizers of finite sets in the case of automorphism groups of countable free-homogeneous structures, Urysohn space and Urysohn sphere, among other related results.

Group topologies on automorphism groups of homogeneous structures

Abstract

We classify all group topologies coarser than the topology of stabilizers of finite sets in the case of automorphism groups of countable free-homogeneous structures, Urysohn space and Urysohn sphere, among other related results.

Paper Structure

This paper contains 27 sections, 72 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Review of some classical constructions of homogeneous structures
  3. Fraïssé construction
  4. Hrushovski's pre-dimension construction
  5. $\omega$-categorical case
  6. A relative minimality criterion for $\tau_{st}$
  7. Minimality and independence
  8. Independence
  9. Simple non-modular Hrushovski structures
  10. Generalized Urysohn spaces
  11. Isometry groups of archimedean Urysohn spaces
  12. Group topologies on $\mathrm{Isom}(\mathcal{U})$ coarser than $\tau_{st}$
  13. Point alignment
  14. Downward lemmas
  15. Proof of Theorem \ref{['3 topologies theorem']}
  16. ...and 12 more sections

Key Result

Theorem A

Let $\mathcal{M}$ be the Fraïssé limit of a free amalgamation class in a countable relational structure. Let $G=\mathrm{Aut}\left(\mathcal{M}\right)$. Then any group topology $\tau\subseteq\tau_{st}$ on $G$ is of the form $\tau_{st}^{X}$, where $X\subseteq M$ is some $G$-invariant set. In particular

Theorems & Definitions (152)

  • Definition 1.1
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Theorem G
  • Lemma 2.1
  • Proposition 2.2
  • ...and 142 more