2025/02/16 by J. de la Nuez González, González, J. de la Nuez, Rob Sullivan +1
Mathematics · #Advanced Topics in Algebra #Advanced Topology and Set Theory #Rings, Modules, and Algebras #math.GR #math.LO #msc:03C15 #msc:05E18 #msc:20B22 #msc:20B27
paper · pdf · doi:10.48550/arxiv.2502.11166
59 pages (this version: presentational changes in intro)
openalex publication_date 2025/02/16 · openalex created_date 2025/10/10 · arxiv created 2026/07/29 · arxiv updated 2026/07/30 · openalex updated_date 2026/07/31
An action U \curvearrowleft G of a group G on a set U is sharply k-transitive if, for any two k-tuples a, b ∈ Uk of distinct elements, there is a unique g ∈ G with a ⋅ g = b. We consider two generalisations of this. Firstly, given Θ≤ \mathbbSk, we define a sharply Θ-transitive action U \curvearrowleft G to be a k-set-transitive action where the restricted action on each k-set of its setwise-stabiliser is isomorphic to the permutation action k \curvearrowleft Θ. An action is sharply \mathbbSk-transitive iff it is sharply k-transitive. We characterise for which Θ≤ \mathbbSk there is a sharply Θ-transitive action on an infinite set, and show that if such an action exists, then the acting group G can be taken to be a finitely generated non-abelian virtually free group. As a consequence, we obtain for k = 2, 3 the first examples of non-split finitely-presented groups admitting sharply k-transitive actions on an infinite set, answering a question of André and Tent, and we obtain a strengthening of the well-known result of Tits that no group admits a sharply k-transitive action on an infinite set for k ≥ 4. Secondly, we generalise sharp k-transitivity to relational structures. Given an action M \curvearrowleft G of a group G on a relational structure M, we say that the action is sharply k-homogeneous if, for any two k-tuples a, b of distinct elements of M where a ↦ b is an isomorphism, there is a unique g ∈ G with a ⋅ g = b. We show that, for 1 ≤ k ≤ 3, a wide range of countable ultrahomogeneous structures admit sharply k-homogeneous actions by finitely generated non-abelian virtually free groups, answering a question of Cameron from 1990.