Geometry & Topology, Vol. 6 (2002) Paper no. 5, pages 91--152.

Convex cocompact subgroups of mapping class groups

Benson Farb, Lee Mosher

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 Teichmuller space. Given a subgroup G of MCG defining an extension L_G:
1---> pi_1(S) ---> L_G ---> G --->1
we prove that if L_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 pi_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.

Keywords. Mapping class group, Schottky subgroup, cocompact subgroup, convexity, pseudo-Anosov

AMS subject classification. Primary: 20F67, 20F65. Secondary: 57M07, 57S25.

DOI: 10.2140/gt.2002.6.91

E-print: arXiv:math.GR/0106190

Submitted to GT on 20 October 2001. Paper accepted 20 February 2002. Paper published 14 March 2002.

Notes on file formats

Benson Farb, Lee Mosher
Department of Mathematics, University of Chicago
5734 University Ave, Chicago, Il 60637, USA
Department of Mathematics and Computer Science
Rutgers University, Newark, NJ 07102, USA


GT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to