On a Theorem of Scott and Swarup
Let $1 \rightarrow H \rightarrow G \rightarrow \mathbb{Z} \rightarrow 1$ be an exact sequence of hyperbolic groups induced by a fully irreducible automorphism $\phi$ of the free group $H$. Let $H_1 (\subset H)$ be a finitely generated distorted subgroup of $G$. Then $H_1$ is of finite index in $H$. This is an analog of a Theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds.