Une action d’un groupe est hautement transitive si agit transitivement sur les -uplets de points distincts pour tout . De nombreux exemples de groupes ayant une action géométrique ou dynamique suffisamment riche admettent des actions hautement transitives. Nous prouvons que si un groupe admet une action hautement transitive telle que ne contient pas le sous-groupe des permutations alternées à support fini, et si est un sous-groupe confiné de , alors l’action de reste hautement transitive en dehors d’un ensemble fini.
Ce résultat fournit un outil pour exclure l’existence d’actions hautement transitives ou classifier les actions hautement transitives de certains groupes. Nous donnons des illustrations concrètes de cela dans le monde des groupes d’origine dynamique. En particulier nous obtenons le premier résultat de classification non triviale des actions hautement transitives d’un groupe de type fini.
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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Mots clés : Highly transitive actions, infinite permutation groups, confined subgroups and uniformly recurrent subgroups, groups of piecewise linear homeomorphisms, Higman–Thompson groups
@article{AHL_2022__5__491_0, author = {Le Boudec, Adrien and Matte Bon, Nicol\'as}, title = {Confined subgroups and high transitivity}, journal = {Annales Henri Lebesgue}, pages = {491--522}, publisher = {\'ENS Rennes}, volume = {5}, year = {2022}, doi = {10.5802/ahl.128}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/ahl.128/} }
Le Boudec, Adrien; Matte Bon, Nicolás. Confined subgroups and high transitivity. Annales Henri Lebesgue, Tome 5 (2022), pp. 491-522. doi : 10.5802/ahl.128. http://archive.numdam.org/articles/10.5802/ahl.128/
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