Convex cocompact subgroups of mapping class groups

Benson Farb, Lee Mosher

Research output: Contribution to journalArticlepeer-review

67 Scopus citations

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 → π1(S) → ΓG → G → 1, we prove that if ΓG 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 π1(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 languageEnglish (US)
Pages (from-to)91-152
Number of pages62
JournalGeometry and Topology
Volume6
DOIs
StatePublished - 2002

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

Keywords

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

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