Confined subgroups and high transitivity

Adrien Le Boudec (1) , Nicolás Matte Bon (2)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 ICJ - Institut Camille Jordan

An action of a group $G$ is highly transitive if $G$ acts transitively on $k$-tuples of distinct points for all $k \geq 1$. Many examples of groups with a rich geometric or dynamical action admit highly transitive actions. We prove that if a group $G$ admits a highly transitive action such that $G$ does not contain the subgroup of finitary alternating permutations, and if $H$ is a confined subgroup of $G$, then the action of $H$ remains highly transitive, possibly after discarding finitely many points. This result provides a tool to rule out the existence of highly transitive actions, and to classify highly transitive actions of a given group. We give concrete illustrations of these applications in the realm of groups of dynamical origin. In particular we obtain the first non-trivial classification of highly transitive actions of a finitely generated group.

Domaines: Mathématiques [math]

Adrien Le Boudec, Nicolás Matte Bon. Confined subgroups and high transitivity. 2021. ⟨hal-03357490v1⟩