Topological properties of Wazewski dendrite groups
Résumé
Homeomorphism groups of generalized Wazewski 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 Wazewski dendrite D is more characteristic than the others because it is the unique one with a dense conjugacy class. For this group G, 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. This allows us to prove that point-stabilizers in G are amenable and to describe the universal Furstenberg boundary of G.
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