The visual boundary of hyperbolic free-by-cyclic groups

Abstract

Let $\varphi$ be an atoroidal outer automorphism of the free group $F_n$. We study the Gromov boundary of the hyperbolic group $G_{\varphi }=F_n{⋊}_{\varphi }\mathbb{Z}$. We explicitly describe a family of embeddings of the complete bipartite graph $K_{3,3}$ into $\partial G_{\varphi }$. To do so, we define the directional Whitehead graph and 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 direct proof of Kapovich-Kleiner's theorem that $\partial G_{\varphi }$ is homeomorphic to the Menger curve if the automorphism is atoroidal and fully irreducible.

25 pages, 3 figures