Cannon–thurston Maps for Hyperbolic Free Group Extensions

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 groups 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 [DT1] 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 [KL5] 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.