The visual boundary of hyperbolic free-by-cyclic groups

Let ϕ be an atoroidal outer automorphism of the free group F_n. We study the Gromov boundary of the hyperbolic group G_ϕ = F_n ⋊_ϕ ℤ.Using the Cannon-Thurston map, we explicitly describe a family of embeddings of the complete bipartite graph K_3,3 into the boundary of the free-by-cyclic group. To do so, we define the directional Whitehead graph and use it to relate the topology of the boundary to the structure of the Rips Machine associated to a fully irreducible outer automorphism of the free group. In particular, we prove that an indecomposable F_n-tree is Levitt type if and only if one of its directional Whitehead graphs contains more than one edge. As an application, we obtain a new proof of Kapovich-Kleiner’s theorem [KK00] that ∂G_ϕ is homeomorphic to the Menger curve if the automorphism is atoroidal and fully irreducible.