Compatible Trees and Outer Automorphisms of a Free Group

The analogy among linear groups, mapping class groups, and outer automorphism groups is imperfect. One point of disanalogy is McCarthy's theorem on two-generator subgroups of mapping class groups. The theorem states that for any two mapping classes, appropriate powers of the two classes generate either a rank two free group or an abelian group. This statement is false for linear groups, and it is unknown whether or not an analogous statement holds for the outer automorphism group. In this work we prove an analogous statement for linearly growing outer automorphisms, developing a general theory of compatibility for R-tree actions along the way.