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The visual boundary of hyperbolic free-by-cyclic groups

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Abstract

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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Authors and Affiliations

  1. Department of Mathematics, University of Haifa, Haifa, 31905, Israel

    Yael Algom-Kfir

  2. Institut de Mathématiques de Marseille Aix Marseille Université, CNRS, Centrale Marseille I2M, Marseille, France

    Arnaud Hilion

  3. Faculty of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel

    Emily Stark

Authors
  1. Yael Algom-Kfir
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  2. Arnaud Hilion
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  3. Emily Stark
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Corresponding author

Correspondence to Yael Algom-Kfir.

Additional information

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

The second author has been supported by the ANR grant DAGGER ANR-16-CE40-0006-01.

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

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Algom-Kfir, Y., Hilion, A. & Stark, E. The visual boundary of hyperbolic free-by-cyclic groups. Isr. J. Math. 244, 501–538 (2021). https://doi.org/10.1007/s11856-021-2191-4

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  • DOI: https://doi.org/10.1007/s11856-021-2191-4

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