Topological properties of Wazewski dendrite groups

Bruno Duchesne (1)

1 IECL - Institut Élie Cartan de Lorraine

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.

Mots clés: Ważewski dendrites; groups of homeomorphisms; Polish groups; Steinhaus property; generic elements; automatic continuity; universal flows

Domaines: Théorie des groupes [math.GR]; Systèmes dynamiques [math.DS]

Bruno Duchesne. Topological properties of Wazewski dendrite groups. Journal de l'École polytechnique — Mathématiques, 2020, 7, pp.431-477. ⟨10.5802/jep.121⟩. ⟨hal-02066642v2⟩