Sharply k-transitive actions on ultrahomogeneous structures

TL;DR

This paper explores sharply k-transitive actions on ultrahomogeneous structures, demonstrating their existence for various structures and k-values, and extends the concept to infinite sets and hyperbolic groups.

Contribution

It introduces the notion of structurally sharply k-transitive actions, providing new examples and answering open questions about their existence on ultrahomogeneous structures and infinite sets.

Findings

Existence of sharply k-transitive actions for k ≤ 3 on countable ultrahomogeneous structures.

Construction of sharply 4 and 5-transitive actions on the random k-hypertournament.

Finitely generated virtually free groups have sharply 2-transitive actions on infinite sets.

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.]