On the geometry of the free factor graph for Aut(F_N) - Bestvina, M; Bridson, M; Wade, R (EMS Press, 2024)
Let Φ be a pseudo-Anosov diffeomorphism of a compact (possibily 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 adb denotes conjugation by b, then the orbits of ⟨ϕ, adb⟩ ∼= Z 2 in the graph of free factors of π1(Σ) are quasiisometrically embedded. It follows that for N ≥ 2 the free factor graph for Aut(FN ) is not hyperbolic, in contrast to the Out(FN ) case.