A finitely generated group with a Cayley graph that is a Gromov hyperbolic space is called a hyperbolic group. Gromov hyperbolic spaces (and the groups) admit a compactification called the Gromov boundary given by the endpoints of geodesic rays. The inclusion of a normal hyperbolic subgroup H of a hyperbolic group G extends continuously to the boundary. This map, known as the Cannon-Thurston map, allows us to determine when the quasiconvex subgroups of H remain quasiconvex in G, in certain cases.
In particular, the speaker will look at the case when H is a finite rank free group and G is a semidirect product of H associated to the action of Z via an automorphism f. The dynamics of f on the Outer space can be used to understand the Cannon-Thurston map of H \to G. The speaker will discuss this and speculate on a generalisation where f is an endomorphism