Two generalisations of sharp k-transitivity

Abstract

An action $U \curvearrowleft G$ of a group $G$ on a set $U$ is sharply $k$-transitive if, for any two $k$-tuples $\bar{a}, \bar{b} \in U^k$ of distinct elements, there is a unique $g \in G$ with $\bar{a} \cdot g = \bar{b}$. We consider two generalisations of this. Firstly, given $Θ\leq \mathbb{S}_k$, 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 $\mathbf{k} \curvearrowleft Θ$. An action is sharply $\mathbb{S}_k$-transitive iff it is sharply $k$-transitive. We characterise for which $Θ\leq \mathbb{S}_k$ 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 $k \geq 4$. Secondly, we generalise sharp $k$-transitivity to relational structures. Given an action $\mathcal{M} \curvearrowleft G$ of a group $G$ on a relational structure $\mathcal{M}$, we say that the action is sharply $k$-homogeneous if, for any two $k$-tuples $\bar{a}, \bar{b}$ of distinct elements of $\mathcal{M}$ where $\bar{a} \mapsto \bar{b}$ is an isomorphism, there is a unique $g \in G$ with $\bar{a} \cdot g = \bar{b}$. We show that, for $1 \leq k \leq 3$, 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.