Article Dans Une Revue Israel Journal of Mathematics Année : 2021

The visual boundary of hyperbolic free-by-cyclic groups

Résumé

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.

Dates et versions

hal-01686944 , version 1 (17-01-2018)

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Yael Algom-Kfir, Arnaud Hilion, Emily Stark. The visual boundary of hyperbolic free-by-cyclic groups. Israel Journal of Mathematics, 2021, 244 (2), pp.501-538. ⟨10.1007/s11856-021-2191-4⟩. ⟨hal-01686944⟩
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