On the geometry of the free factor graph for Aut$(F_N)$

Abstract

Let $\Phi$ be a pseudo-Anosov diffeomorphism of a compact (possibly non-orientable) surface $\Sigma$ with one boundary component. We show that if $b\in {\pi }_1(\Sigma )$ is the boundary word, $\varphi \in \mathrm{Aut}({\pi }_1(\Sigma ))$ is a representative of $\Phi$ fixing $b$, and ${\mathrm{ad}}_b$ denotes conjugation by $b$, then the orbits of $⟨\varphi ,{\mathrm{ad}}_b⟩\cong {\mathbb{Z}}^2$ in the graph of free factors of ${\pi }_1(\Sigma )$ are quasi-isometrically embedded. It follows that for $N\ge 2$ the free factor graph for $\mathrm{Aut}(F_N)$ is not hyperbolic, in contrast to the $\mathrm{Out}(F_N)$ case.