Rob Sullivan (Charles University, Prague) – Sharply $k$-homogeneous actions on Fraïssé structures

Notes: unusual room and time.

Given an action of a group $G$ on a relational Fraïssé structure $M$, we call this action sharply $k$-homogeneous if, for each isomorphism $f : A \to B$ 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 $k \leq 3$ 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.