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.
[Cha12] Properties of hyperbolic groups: Free normal subgroups, quasiconvex subgroups and actions of maximal growth, Ph. D. Thesis, Vanderbilt University, Nashville, USA (2012) | MR
[FLMMS20] High transitivity for more groups acting on trees (2020) (https://arxiv.org/abs/2003.11116)
[GGS19] Maximal subgroups of countable groups, a survey (2019) (https://arxiv.org/abs/1909.09361)
[GNS00] Automata, dynamical systems, and groups, Tr. Mat. Inst. Steklova, Volume 231 (2000), pp. 134-214 | Zbl
[GW15] Uniformly recurrent subgroups, Recent trends in ergodic theory and dynamical systems. International conference in honor of S. G. Dani’s 65th birthday, Vadodara, India, December 26–29, 2012 (Bhattacharya, Siddhartha et al., eds.) (Contemporary Mathematics), Volume 631, American Mathematical Society, 2015, pp. 63-75 | MR | Zbl
[Hig74] Finitely presented infinite simple groups, Notes on Pure Mathematics, 8, Department of Mathematics, Australian National University, Canberra, 1974 | Zbl
[LBMB19] Triple transitivity and non-free actions in dimension one (2019) (https://arxiv.org/abs/1906.05744)
[LBMB20] A commutator lemma for confined subgroups and applications to groups acting on rooted trees (2020) (https://arxiv.org/abs/2006.08677)
[MB18] Rigidity properties of topological full groups of pseudogroups over the Cantor set (2018) (https://arxiv.org/abs/1801.10133)
[Zal95] Group rings of simple locally finite groups, Finite and locally finite groups. Proceedings of the NATO Advanced Study Institute held in Istanbul, Turkey, 14-27 August 1994 (NATO ASI Series. Series C. Mathematical and Physical Sciences), Volume 471, Kluwer Academic Publishers, 1995, pp. 219-246 | MR | Zbl