Pseudo-Anosov subgroups of fibered 3-manifold groups

Abstract

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

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

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

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 ${\Gamma }_{\Theta }$.