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 $D_{\infty }$ 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 $G_{\infty }$.

An erratum for Theorem 1.12 has appeared in Journal de l'Ecole polytechnique - Mathématiques, Tome 11 (2024), pp. 1029-1034. This does not affect the other statements