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.Using the Cannon-Thurston map, we explicitly describe a family of embeddings of the complete bipartite graph K3,3 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 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 new proof of Kapovich-Kleiner’s theorem [KK00] that ∂Gϕ is homeomorphic to the Menger curve if the automorphism is atoroidal and fully irreducible.
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© 2021, The Hebrew University of Jerusalem.
ASJC Scopus subject areas
- General Mathematics