On the geometry of the free factor graph for Aut(FN)

Let Φ be a pseudo-Anosov diffeomorphism of a compact (possibly non-orientable) surface Σ with one boundary component. We show that if b ∈ π₁(Σ) is the boundary word, φ ∈ Aut(π₁(Σ)) is a representative of Φ fixing b , and {ad}b denotes conjugation by b , then the orbits of φ,{ad}b Z² in the graph of free factors of π₁(Σ) are quasi-isometrically embedded. It follows that for N ≥ 2 the free factor graph for Aut(FN) is not hyperbolic, in contrast to the Out(FN) case.