## Abstract

We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmüller space. Given a subgroup G of MCG defining an extension 1 → π_{1}(S) → Γ_{G} → G → 1, we prove that if Γ_{G} is a word hyperbolic group then G is a convex cocompact subgroup of MCG. When G is free and convex cocompact, called a Schottky subgroup of MCG, the converse is true as well; a semidirect product of π_{1}(S) by a free group G is therefore word hyperbolic if and only if G is a Schottky subgroup of MCG. The special case when G = Z follows from Thurston's hyperbolization theorem. Schottky subgroups exist in abundance: sufficiently high powers of any independent set of pseudo-Anosov mapping classes freely generate a Schottky subgroup.

Original language | English (US) |
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Pages (from-to) | 91-152 |

Number of pages | 62 |

Journal | Geometry and Topology |

Volume | 6 |

DOIs | |

State | Published - 2002 |

## All Science Journal Classification (ASJC) codes

- Geometry and Topology

## Keywords

- Cocompact subgroup
- Convexity
- Mapping class group
- Pseudo-anosov
- Schottky subgroup