2020/12/07 by Adrien Le Boudec, Nicolás Matte Bon, Boudec, Adrien Le +1
Mathematics · #Geometric and Algebraic Topology #Advanced Topology and Set Theory #Finite Group Theory Research
paper · doi:10.48550/arxiv.2012.03997
An action of a group G is highly transitive if G acts transitively on k-tuples of distinct points for all k ≥ 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.