Uniform independence for Dehn twist automorphisms of a free group

TL;DR

This paper proves a uniform independence criterion for Dehn twist automorphisms of free groups, extending McCarthy's theorem from surface mapping class groups to free group automorphisms.

Contribution

It establishes a uniform power bound for independence of Dehn twist automorphisms in free groups, analogous to known results for mapping class groups.

Findings

Proves an independence dichotomy for linearly growing outer automorphisms of free groups.

Provides a geometric criterion for when automorphisms generate a free group or are abelian.

Establishes a uniform power bound depending only on the rank of the 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 $\langle \sigma^N,\tau^N\rangle$ 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 $\sigma$ 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.