The visual boundary of hyperbolic free-by-cyclic groups

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Let ϕ be an atoroidal outer automorphism of the free group Fₙ. We study the Gromov boundary of the hyperbolic group Gϕ = Fₙ ϕ Z. We explicitly describe a family of embeddings of the complete bipartite graph K₃,₃ into ∂ Gϕ. To do so, we define the directional Whitehead graph and prove that an indecomposable Fₙ-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 direct proof of Kapovich-Kleiner's theorem that ∂ Gϕ is homeomorphic to the Menger curve if the automorphism is atoroidal and fully irreducible.