• DOI:10.1007/s11856-016-1426-2
• Corpus ID: 9898253

Cannon–Thurston maps for hyperbolic free group extensions

@article{Dowdall2015CannonThurstonMF,
  title={Cannon–Thurston maps for hyperbolic free group extensions},
  author={Spencer Dowdall and Ilya Kapovich and Samuel J. Taylor},
  journal={Israel Journal of Mathematics},
  year={2015},
  volume={216},
  pages={753-797}
}

• S. Dowdall, Ilya Kapovich, Samuel J. Taylor
• Published 23 June 2015
• Mathematics
• Israel Journal of Mathematics

This paper gives a detailed analysis of the Cannon–Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free group of finite rank at least 3 and consider a convex cocompact subgroup Γ ≤ Out(F), i.e. one for which the orbit map from Γ into the free factor complex of F is a quasi-isometric embedding. The subgroup Γ determines an extension EΓ of F, and the main theorem of Dowdall–Taylor [DT14] states that in this situation EΓ is hyperbolic if and only if Γ is purely atoroidal.Here, we give an explicit geometric description of the Cannon–Thurston maps ∂F → ∂EΓ for these hyperbolic free group extensions, the existence of which follows from a general result of Mitra. In particular, we obtain a uniform bound on the multiplicity of the Cannon–Thurston map, showing that this map has multiplicity at most 2 rank(F). This theorem generalizes the main result of Kapovich and Lustig [KL15] which treats the special case where Γ is infinite cyclic. We also answer a question of Mahan Mitra by producing an explicit example of a hyperbolic free group extension for which the natural map from the boundary of Γ to the space of laminations of the free group (with the Chabauty topology) is not continuous.