  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-Mosher and Farb.
