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


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


An action of a group G is highly transitive if G acts transitively on k-tuples of distinct points for all k1. 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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