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Topological properties of Wazewski dendrite groups

Bruno Duchesne

Abstract

Homeomorphism groups of generalized Ważewski dendrites act on the infinite countable set of branch points of the dendrite and thus have a nice Polish topology. In this paper, we study them in the light of this Polish topology. The group of the universal Ważewski dendrite $D_\infty$ is more characteristic than the others because it is the unique one with a dense conjugacy class. For this group $G_\infty$, we show some of its topological properties like existence of a comeager conjugacy class, the Steinhaus property, automatic continuity and the small index subgroup property. Moreover, we identify the universal minimal flow of $G_\infty$. This allows us to prove that point-stabilizers in $G_\infty$ are amenable and to describe the universal Furstenberg boundary of $G_\infty$.

Topological properties of Wazewski dendrite groups

Abstract

Homeomorphism groups of generalized Ważewski dendrites act on the infinite countable set of branch points of the dendrite and thus have a nice Polish topology. In this paper, we study them in the light of this Polish topology. The group of the universal Ważewski dendrite is more characteristic than the others because it is the unique one with a dense conjugacy class. For this group , we show some of its topological properties like existence of a comeager conjugacy class, the Steinhaus property, automatic continuity and the small index subgroup property. Moreover, we identify the universal minimal flow of . This allows us to prove that point-stabilizers in are amenable and to describe the universal Furstenberg boundary of .

Paper Structure

This paper contains 19 sections, 60 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Generic elements
  3. Automatic continuity
  4. Small index property
  5. Universal minimal flows
  6. Ważewski dendrites and their homeomorphism groups
  7. Ważewski dendrites
  8. Dynamics of individual elements
  9. Fraïssé theory and generic elements
  10. Existence of a dense conjugacy class
  11. Existence of a comeager conjugacy class
  12. Automatic continuity
  13. Universal minimality of the topology
  14. Small index subgroups
  15. Universal minimal flow
  16. ...and 4 more sections

Key Result

Proposition 1.1

The Polish group $G_S$ has a dense conjugacy class if and only if $S=\{\infty\}$.

Figures (2)

  • Figure 1: The Julia set of $z\mapsto z^2+i$. Picture realized with Mathematica.
  • Figure 2: The simple dendrite $D$ with a converging but not convex linear order $\prec$.

Theorems & Definitions (126)

  • Proposition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Proposition 1.9
  • Theorem 1.10
  • ...and 116 more