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 $F_N$ denote a free groups of finite rank $N\ge 3$ and consider a \emph{convex cocompact} subgroup $\Gamma\le Out(F_N)$, i.e. one for which the orbit map from $\Gamma$ into the free factor complex of $F_N$ is a quasi-isometric embedding. The subgroup $\Gamma$ determines an extension $E_\Gamma$ of $F_N$, and the main theorem of Dowdall--Taylor \cite{DT1} states that in this situation $E_\Gamma$ is hyperbolic if and only if $\Gamma$ is purely atoroidal. Here, we give an explicit geometric description of the Cannon--Thurston maps $\partial F_N\to\partial E_\Gamma$ 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 $2N$. This theorem generalizes the main result of Kapovich and Lustig \cite{KapLusCT} which treats the special case where $\Gamma$ 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 $\Gamma$ to the space of laminations of the free group (with the Chabauty topology) is not continuous.