## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 1247-1282 |

Number of pages | 36 |

Journal | Groups, Geometry, and Dynamics |

Volume | 8 |

Issue number | 4 |

DOIs | |

State | Published - 2014 |

Externally published | Yes |

## Keywords

- Convex cocompact
- Fibered 3-manifold
- Gromov hyperbolic
- Hyperbolic
- Mapping class group
- Pseudo-Anosov

## ASJC Scopus subject areas

- Geometry and Topology
- Discrete Mathematics and Combinatorics