  A Kleinian group $\Gamma < \mathrm{Isom}(\mathbb H^3)$ is called convex
cocompact if any orbit of $\Gamma$ in $\mathbb H^3$ is quasiconvex or,
equivalently, $\Gamma$ acts cocompactly on the convex hull of its limit set in
$\partial \mathbb H^3$. Subgroup stability is a strong quasiconvexity condition
in finitely generated groups which is intrinsic to the geometry of the ambient
group and generalizes the classical quasiconvexity condition above.
Importantly, it coincides with quasiconvexity in hyperbolic groups and convex
cocompactness in mapping class groups. Using the Morse boundary, we develop an
equivalent characterization of subgroup stability which generalizes the above
boundary characterization from Kleinian groups.
