#### 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

and

Department of Mathematics and Computer Science

Rutgers University, Newark, NJ 07102, USA

Email: farb@math.uchicago.edu, mosher@andromeda.rutgers.edu

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