  For a finitely generated group, there are two recent generalizations of the
notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable
subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor
defined stability and proved stability is equivalent to convex cocompactness in
mapping class groups. Another natural generalization of quasiconvexity is given
by the notion of a Morse or strongly quasiconvex subgroup of a finitely
generated group, studied recently by Tran and Genevois. In general, a subgroup
is stable if and only if the subgroup is Morse and hyperbolic. In this paper,
we prove that two properties of being Morse and stable coincide for a subgroup
of infinite index in the mapping class group of an oriented, connected, finite
type surface with negative Euler characteristic.
