A simple acylindrical recipe for non-split characteristic $2$ sharply $k$-transitive actions and their generalizations

Abstract

We provide a fairly simple list of conditions on a group $G$ acting acylindrically on a $\delta$-hyperbolic metric space sufficient for the group to admit an action on a set that is $k$-sharp, transitive on $k$-sets and has the property that for any $k$-set $A$ the setwise stabilizer of $A$ acts on $A$ as prescribed by some fixed subgroup of the group of permutations of $k$ elements. In particular, this yields a large class of non-split sharply $2$ and $3$-transitive actions containing many finitely presented groups, and allows us to characterize sharp $2$-transitivity within the class of hyperbolic groups that do not contain a copy of $C_2\times C_2$.