Highly transitive actions of $\operatorname{Out}(F_n)$

Abstract

An action of a group on a set is called $k$-transitive if it is transitive on ordered $k$-tuples and highly transitive if it is $k$-transitive for every $k$. We show that for $n \ge 4$ the group $\operatorname{Out}(F_n) = \operatorname{Aut}(F_n) / \mathrm{Inn}(F_n)$ admits a faithful highly transitive action on a countable set.