On the geometry of the free factor graph for ${\rm{Aut}}(F_N)$

Abstract

Let $Φ$ be a pseudo-Anosov diffeomorphism of a compact (possibly non-orientable) surface $Σ$ with one boundary component. We show that if $b \in π_1(Σ)$ is the boundary word, $ϕ\in {\rm{Aut}}(π_1(Σ))$ is a representative of $Φ$ fixing $b$, and ${\rm{ad}}_b$ denotes conjugation by $b$, then the orbits of $\langle ϕ, {\rm{ad}}_b \rangle\cong\mathbb{Z}^2$ in the graph of free factors of $π_1(Σ)$ are quasi-isometrically embedded. It follows that for $N \geq 2$ the free factor graph for ${\rm{Aut}}(F_N)$ is not hyperbolic, in contrast to the ${\rm{Out}}(F_N)$ case.