Sharply k-homogeneous actions on Fraïssé structures

Given an action of a group G on a relational Fraïssé structure M, we call this action *sharply k-homogeneous* if, for each isomorphism f : A -> B of substructures of M of size k, there is exactly one element of G whose action extends f. This generalises the well-known notion of a sharply k-transitive action on a set, and was previously investigated by Cameron, Macpherson and Cherlin. I will discuss recent results with J. de la Nuez González which show that a wide variety of Fraïssé structures admit sharply k-homogeneous actions for k ≤ 3 by finitely generated virtually free groups. Our results also specialise to the case of sets, giving the first examples of finitely presented non-split infinite groups with sharply 2-transitive/sharply 3-transitive actions.