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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Convex cocompactness and stability in mapping class groups

Matthew Gentry Durham and Samuel J Taylor

Algebraic & Geometric Topology 15 (2015) 2837–2857
Abstract

We introduce a strong notion of quasiconvexity in finitely generated groups, which we call stability. Stability agrees with quasiconvexity in hyperbolic groups and is preserved under quasi-isometry for finitely generated groups. We show that the stable subgroups of mapping class groups are precisely the convex cocompact subgroups. This generalizes a well-known result of Behrstock and is related to questions asked by Farb and Mosher and by Farb.

Keywords
convex cocompact subgroups of mapping class groups, stability, quasiconvexity, hyperbolic groups
Mathematical Subject Classification 2010
Primary: 20F65, 51H05
Secondary: 57M07, 30F60
References
Publication
Received: 2 July 2014
Revised: 1 January 2015
Accepted: 12 March 2015
Published: 10 December 2015
Authors
Matthew Gentry Durham
Department of Mathematics
University of Michigan
3079 East Hall
530 Church St.
Ann Arbor, MI 48109
USA
http://www-personal.umich.edu/~durhamma/
Samuel J Taylor
Department of Mathematics
Yale University
10 Hillhouse Ave.
New Haven, CT 06520
USA
http://users.math.yale.edu/users/taylor/