Uniform independence for Dehn twist automorphisms of a free group

Abstract

McCarthy's Theorem for the mapping class group of a closed hyperbolic surface states that for any two mapping classes $\sigma ,\tau \in \mathrm{Mod}(S)$ there is some power $N$ such that the group $⟨{\sigma }^N,{\tau }^N⟩$ 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 $\sigma$ and $\tau$ 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 \emph{uniform}: $N=4$ suffices. In a similar style, we obtain an $N$ that depends only on the rank of the free group.

arXiv comment39 pages, 3 figures, results part of the author's Ph. D. thesis; revised to add examples and clarify at the request of a referee. Accepted version, Proc. Lond. Math. Soc

References in corpus (8)

• Core and intersection number for group actions on trees
• Geometric Intersection Number and analogues of the Curve Complex for free groups
• Ping-pong and Outer space
• Dehn twists have roots
• Growth of intersection numbers for free group automorphisms
• Sur les Automorphismes de Groupes Libres et de Groupes de Surface
• Fully irreducible Automorphisms of the Free Group via Dehn twisting in ${♯}_k(S^2\times S^1)$
• Length Function Compatibility for Group Actions on Real Trees