Pseudo-Anosov subgroups of fibered 3-manifold groups

Let $S$ be a hyperbolic surface and let $\mathring{S}$ be the surface obtained from $S$ by removing a point. The mapping class groups $\mathrm {Mod}(S)$ and $\mathrm {Mod}(\mathring{S})$ fit into a short exact sequence \[ 1 \to \pi_1(S) \to \mathrm {Mod}(\mathring{S}) \to \mathrm {Mod}(S) \to 1. \] If $M$ is a hyperbolic $3$-manifold that fibers over the circle with fiber $S$, then its fundamental group fits into a short exact sequence \[ 1 \to \pi_1(S) \to \pi_1(M) \to \mathbb Z \to 1 \] that injects into the one above. We show that, when viewed as subgroups of $\mathrm {Mod} (\mathring{S})$, finitely generated purely pseudo-Anosov subgroups of $\pi_1(M)$ are convex cocompact in the sense of Farb and Mosher. More generally, if we have a $\delta$-hyperbolic surface group extension \[ 1 \to \pi_1(S) \to \Gamma_\Theta \to \Theta \to 1, \] any quasiisometrically embedded purely pseudo-Anosov subgroup of $\Gamma_\Theta$ is convex cocompact in $\mathrm {Mod}(\mathring{S})$. We also obtain a generalization of a theorem of Scott and Swarup by showing that finitely generated subgroups of $\pi_1(S)$ are quasiisometrically embedded in hyperbolic extensions $\Gamma_\Theta$.