In his proof of the Tits alternative for the mapping class group of a surface, McCarthy also proved that given any two mapping classes σ and τ, there exists an integer N such that the group generated by ⟨ σN, τN ⟩ is either free of rank two or abelian. An analogous statement for two-generator subgroups of a linear group is false, due to the presence of the Heisenberg group. In the setting of Out(Fn), whether or not such a statement is true remains open, though there are many partial results. In the first half-hour of this talk I will give an overview the analogy among the three families of groups, and in this context survey previous work on the problem; in the second half-hour I will give a more detailed introduction to the Guirardel core and other topological models built on Bass–Serre theory, which play a role (somewhat) analogous to curves and surfaces in the study of Out(Fn); in the last half-hour I will illustrate ideas in my proof of a McCarthy-type theorem for linearly growing outer automorphisms with examples, and conclude by discussing the relationship between this work and questions about growth rates, again drawing on the analogy with linear groups and mapping class groups.