## Abstract

Let ϕ be an atoroidal outer automorphism of the free group F_{n}. We study the Gromov boundary of the hyperbolic group G_{ϕ} = F_{n} ⋊_{ϕ} ℤ.Using the Cannon-Thurston map, we explicitly describe a family of embeddings of the complete bipartite graph K_{3,3} 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 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 new proof of Kapovich-Kleiner’s theorem [KK00] that ∂G_{ϕ} is homeomorphic to the Menger curve if the automorphism is atoroidal and fully irreducible.

Original language | English |
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Journal | Israel Journal of Mathematics |

DOIs | |

State | Accepted/In press - 2021 |

### Bibliographical note

Funding Information:The first and third authors were supported by ISF grant 1941/14.

Funding Information:

The third author was supported by the Azrieli Foundation and was partially supported at the Technion by a Zuckerman STEM Leadership Postdoctoral Fellowship.

Publisher Copyright:

© 2021, The Hebrew University of Jerusalem.

## ASJC Scopus subject areas

- Mathematics (all)