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 $G_\infty$.
Highlights
A dendrite is a continuum that is locally connected and such that any two points are connected by a unique arc
Groups acting by homeomorphisms on dendrites share some properties with groups acting by isometries on R-trees but some dendrite group properties are very far from properties of groups acting by isometries on R-trees, for example some have the fixed-point property for isometric actions on Hilbert spaces (the so-called property (FH))
Generic elements. — The aim of this paper is to study some topological properties of the Polish group GS
Summary
A dendrite is a continuum (i.e., a connected metrizable compact space) that is locally connected and such that any two points are connected by a unique arc (see [Nad92] for background on continua and dendrites). It is well known that automatic continuity implies uniqueness of the Polish group topology We have another proof of a particular case of a result due to Kallman. It is known that an amenable group acting continuously on a dendrite stabilizes a subset with at most two points [SY17] This amenability result enables us to identify the universal Furstenberg boundary of G∞, that is, the universal strongly proximal minimal G∞-flow. Even if the set of end points is a dense Gδ-orbit in D∞, the natural map G∞/Gξ → D∞ is not a homeomorphism (Proposition 8.10) and D∞ is a Furstenberg boundary of G∞ but not the universal one. At the end of this paper, we give another description of this universal Furstenberg boundary It appears as a closed subset of a natural countable product of totally disconnected compact spaces.
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