Uniform independence in linear groups

Uniform independence for Dehn twist automorphisms of a free group

McCarthy's Theorem for the mapping class group of a closed hyperbolic surface states that for any two mapping classes $σ,τ\in \mathrm{Mod}(S)$ there is some power $N$ such that the group $\langle σ^N,τ^N\rangle$ is either free of rank two or abelian, and gives a geometric criterion for the dichotomy. The analogous statement is false in linear groups, and unresolved for outer automorphisms of a free group. Several analogs are known for exponentially growing outer automorphisms satisfying various technical hypothesis. In this article we prove an analogous statement when $σ$ and $τ$ are linearly …