The visual boundary of hyperbolic free-by-cyclic groups

TL;DR

This paper investigates the structure of the Gromov boundary of hyperbolic free-by-cyclic groups, providing explicit embeddings of complex graphs and characterizing boundary homeomorphisms to the Menger curve.

Contribution

It introduces the directional Whitehead graph, characterizes Levitt type trees, and offers a new proof that the boundary is homeomorphic to the Menger curve under certain conditions.

Findings

Explicit embeddings of K_{3,3} into the boundary

Characterization of Levitt type trees via Whitehead graphs

Proof that the boundary is homeomorphic to the Menger curve

Abstract

Let $\phi$ be an atoroidal outer automorphism of the free group $F_n$. We study the Gromov boundary of the hyperbolic group $G_{\phi} = F_n \rtimes_{\phi} \mathbb{Z}$. We explicitly describe a family of embeddings of the complete bipartite graph $K_{3,3}$ into $\partial G_\phi$. 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 $\partial G_\phi$ is homeomorphic to the Menger curve if the automorphism is atoroidal and fully irreducible.