Pseudo-Anosov subgroups of fibered 3-manifold groups

Abstract

Let S be a hyperbolic surface and let ˚ S be the surface obtained from S by 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 fits into a short exact sequence 1 → π1(S) → π1(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 δ ‐hyperbolic surface group extension 1 → π1(S) → ΓΘ → Θ → 1, any quasiisometrically embedded purely pseudo-Anosov subgroup of ΓΘ is convex cocompact in Mod( ˚ S). We also obtain a generalization of a theorem of Scott and Swarup by showing that finitely gener ated subgroups of π1(S) are quasiisometrically embedded in hyperbolic extensions ΓΘ.