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McCarthy's theorem for the mapping class group of a closed hyperbolic surface states that for any two mapping classes    σ  ,  τ  ∈   Mod   (  S  )    there is some power   N   such that the group    ⟨   σ  N   ,   τ  N   ⟩    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 growing outer automorphisms of    F  r   , and give a geometric criterion for the dichotomy. Further, Hamidi‐Tehrani proved that for Dehn twists in the mapping class group this independence dichotomy is  uniform :    N  =  4    suffices. In a similar style, we obtain an   N   that depends only on the rank of the free group.

Uniform independence for Dehn twist automorphisms of a free group