TY - JOUR

T1 - Cannon–Thurston maps for hyperbolic free group extensions

AU - Dowdall, Spencer

AU - Kapovich, Ilya

AU - Taylor, Samuel J.

N1 - Publisher Copyright:
© 2016, Hebrew University of Jerusalem.

PY - 2016/10/1

Y1 - 2016/10/1

N2 - 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 EΓof F, and the main theorem of Dowdall–Taylor [DT14] states that in this situation EΓis hyperbolic if and only if Γ is purely atoroidal. Here, we give an explicit geometric description of the Cannon–Thurston maps ∂F → ∂EΓfor 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.

AB - 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 EΓof F, and the main theorem of Dowdall–Taylor [DT14] states that in this situation EΓis hyperbolic if and only if Γ is purely atoroidal. Here, we give an explicit geometric description of the Cannon–Thurston maps ∂F → ∂EΓfor 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.

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U2 - 10.1007/s11856-016-1426-2

DO - 10.1007/s11856-016-1426-2

M3 - Article

AN - SCOPUS:84991442264

SN - 0021-2172

VL - 216

SP - 753

EP - 797

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 2

ER -