Convex cocompact subgroups of mapping class groups

We develop a theory of convex cocompact subgroups of the mapping class group $M C G$ of a closed, oriented surface $S$ of genus at least 2, in terms of the action on Teichmüller space. Given a subgroup $G$ of $M C G$ defining an extension $1 \to π_{1} (S) \to Γ_{G} \to G \to 1$ , we prove that if $Γ_{G}$ is a word hyperbolic group then $G$ is a convex cocompact subgroup of $M C G$ . When $G$ is free and convex cocompact, it is called a Schottky subgroup.

Keywords

mapping class group, Schottky subgroup, cocompact subgroup, convexity, pseudo-Anosov

Mathematical Subject Classification 2000

Primary: 20F67, 20F65

Secondary: 57M07, 57S25

References

Forward citations

Publication

Received: 20 October 2001

Accepted: 20 February 2002

Published: 14 March 2002

Proposed: Walter Neumann

Seconded: Shigeyuki Morita, Robion Kirby

Authors

Benson Farb
Department of Mathematics University of Chicago 5734 University Ave Chicago Illinois 60637 USA

Lee Mosher
Department of Mathematics and Computer Science Rutgers University Newark New Jersey 07102 USA

Volume 6, issue 1 (2002)

Benson Farb and Lee Mosher