Sharply k-transitive actions on ultrahomogeneous structures
J. de la Nuez González, Rob Sullivan
TL;DR
The paper addresses when relational Fraïssé structures $\mathcal{M}$ admit structurally sharply $k$-transitive actions by groups acting via automorphisms. It develops a seed/completion framework guided by stationary weak independence relations (SWIR) and shows, for $k\le3$, that many countable ultrahomogeneous Fraïssé structures admit sharply $k$-transitive actions by finitely generated non-abelian free groups, with broader cases by virtually free groups; it also constructs sharply $k$-transitive actions for the random $k$-hypertournament ($3\le k\le5$) and for reducts of $\mathbb{Q}$, thereby connecting to classical nonexistence results for $k\ge4$. The robustness framework identifies robust subgroups (e.g., $A_4$, $A_5$) that guarantee sharply $\Theta$-transitive actions by finitely generated virtually free groups and yields a spectrum of explicit examples across $\mathcal{M}$ (dense linear orders, betweenness, circular order, separation structures, and $k$-tournaments). The work combines combinatorial group theory, model theory of Fraïssé limits, and independence relations to produce generic, large-action spaces and poses several natural open questions guiding future exploration of structurally sharp transitivity on broader relational structures.
Abstract
Given an action of a group $G$ by automorphisms on an infinite relational structure $\mathcal{M}$, we say that the action is structurally sharply $k$-transitive if, for any two $k$-tuples $\bar{a}, \bar{b} \in M^k$ of distinct elements such that $\bar{a} \mapsto \bar{b}$ is an isomorphism, there exists exactly one element of $G$ sending $\bar{a}$ to $\bar{b}$. This generalises the well-known notion of a sharply $k$-transitive action on a set. We show that, for $k \leq 3$, a wide range of countable ultrahomogeneous structures admit structurally sharply $k$-transitive actions by finitely generated virtually free groups, giving a substantial answer to a question of Cameron from the book Oligomorphic Permutation Groups. We also show that the random $k$-hypertournament admits a structurally sharply $k$-transitive action for $k=4,5$, and that $\mathbb{Q}$ and several of its reducts admit structurally sharply $k$-transitive actions for all $k$. (This contrasts with the case of sets, where for $k \geq 4$ there are no sharply $k$-transitive actions on infinite sets by results of Tits and Hall.) We also show the existence of sharply $2$-transitive actions of finitely generated virtually free groups on an infinite set, solving the open question of whether such actions exist for hyperbolic groups. [Note: this is an early working draft.]