An action of a group is highly transitive if acts transitively on -tuples of distinct points for all . Many examples of groups with a rich geometric or dynamical action admit highly transitive actions. We prove that if a group admits a highly transitive action such that does not contain the subgroup of finitary alternating permutations, and if is a confined subgroup of , then the action of 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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