# Topological properties of Wazewski dendrite groups

## TL;DR

This paper investigates the topological and dynamical properties of homeomorphism groups of Wazewski dendrites, especially the universal one, revealing their unique features and minimal flows.

## Contribution

It characterizes the universal Wazewski dendrite group, showing properties like a dense conjugacy class, automatic continuity, and identifying its universal minimal flow.

## Key findings

- The universal Wazewski dendrite group has a dense conjugacy class.
- It possesses the Steinhaus property and automatic continuity.
- The universal minimal flow and Furstenberg boundary are explicitly described.

## Abstract

Homeomorphism groups of generalized Wa\.zewski 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 Wa\.zewski 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 $G_\infty$.