Seminar

Let $F_N$ be a free group of finite rank $N\ge 3$ and let $\Gamma\le Out(F_N)$ be a finitely generated "convex cocompact" subgroup, that is, such that the orbit map from $\Gamma$ to the free factor complex of $F_N$ is a quasi-isometric embedding. Assume also that $\Gamma$ is purely atoroidal. In this case $\Gamma$ determines an extension group $E_\Gamma$ of $\Gamma$ with the quotient $E_\Gamma/F_N=\Gamma$, and it is known by a result of Dowdall and Taylor that the group $E_\Gamma$ is then word-hyperbolic. By a general result of Mitra the inclusion of $F_N$ in $E_\Gamma$ extends to a continuous surjective $F_N$-equivariant map between their hyperbolic boundaries $j: \partial F_N\to \partial E_\Gamma$, called the Cannon-Thurston map. We analyze the structure of this map and prove that the map is finite-to-one, with multiplicity at most $2N$. The talk is based on a joint paper with Spencer Dowdall and Sam Taylor.