Cannon–Thurston maps for hyperbolic free group extensions

Abstract

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 nite rank at least 3 and consider a convex cocompact subgroup Out(F), i.e. one for which the orbit map from into the free factor complex of F is a quasi-isometric embedding. The subgroup determines an extension E of F, and the main theorem of Dowdall{Taylor (DT1) states that in this situation E is hyperbolic if and only if is purely atoroidal. Here, we give an explicit geometric description of the Cannon{Thurston maps @F! @E 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 is innite 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 to the space of laminations of the free group (with the Chabauty topology) is not continuous.