Confined subgroups and high transitivity
Annales Henri Lebesgue, Volume 5 (2022), pp. 491-522.

KeywordsHighly transitive actions, infinite permutation groups, confined subgroups and uniformly recurrent subgroups, groups of piecewise linear homeomorphisms, Higman–Thompson groups

### Abstract

An action of a group $G$ is highly transitive if $G$ acts transitively on $k$-tuples of distinct points for all $k\ge 1$. 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.

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