Theti Pseudo-Anosov subgroups of fibered 3-manifold groups

Let S be a hyperbolic surface and let S be the surface obtained from S by o removing a point. The mapping class groups Mod(S) and Mod(S) fit into a short exact sequence 1→π1 (S) → Mod(S) → Mod(S) → 1. If M is a hyperbolic 3-manifold that fibers over the circle with fiber S, then its fundamental group its into a short exact sequence 1→π 1 (S)→π 1(M) → Z 1 that injects into the one above. We show that, when viewed as subgroups of Mod(S), finitely generated purely pseudo-Anosov subgroups of π1(M) are convex cocompact in the sense of Farb and Mosher. More generally, if we have a 1-hyperbolic surface group extension 1 → π1 (S) → ⌈φ → φ 1, any quasiisometrically embedded purely pseudo-Anosov subgroup of ⌈φ is convex coo compact in Mod(S). We also obtain a generalization of a theorem of Scott and Swarup by showing that finitely generated subgroups of π1(S) are quasiisometrically embedded in hyperbolic extensions ⌈φ.