Topological properties of Ważewski dendrite groups

Duchesne, Bruno

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 431-477.

Les groupes d’homéomorphismes des dendrites de Ważewski généralisées agissent sur l’ensemble des points de branchement de la dendrite et possèdent ainsi une topologie de groupe polonais agréable. Dans cet article, nous étudions ces groupes à la lumière de cette topologie polonaise. Le groupe d’homéomorphismes de la dendrite universelle de Ważewski $D_{\infty }$ est remarquable puisque c’est le seul avec une classe de conjugaison dense. Pour ce groupe, $G_{\infty }$, nous explorons et prouvons certaines de ses propriétés topologiques comme l’existence d’une classe de conjugaison comaigre, la propriété de Steinhaus, la propriété de continuité automatique, la propriété des groupes de petit indice et une caractérisation de la topologie. De plus, nous décrivons le flot minimal universel de $G_{\infty }$ et des stabilisateurs de points de $D_{\infty }$. Cela nous permet de montrer que les stabilisateurs de points de $D_{\infty }$ sont des groupes moyennables et de donner une description simple et explicite du bord de Furstenberg universel de $G_{\infty }$.

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 explore and prove some of its topological properties like the existence of a comeager conjugacy class, the Steinhaus property, automatic continuity, the small index subgroup property and characterization of the topology. Moreover, we describe the universal minimal flow of $G_{\infty }$ and of point-stabilizers. This enables us to prove that point-stabilizers in $G_{\infty }$ are amenable and to give a simple and completely explicit description of the universal Furstenberg boundary of $G_{\infty }$.

Classification : 22F50, 57S05, 37B05
Keywords: Ważewski dendrites, groups of homeomorphisms, Polish groups, Steinhaus property, generic elements, automatic continuity, universal flows
Mot clés : Dendrites de Ważewski, groupes d’homéomorphismes, groupes polonais, propriété de Steinhaus, éléments génériques, continuité automatique, flots universels

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