  We show the mapping class group, CAT(0) groups, the fundamental groups of
closed 3-manifolds, and certain relatively hyperbolic groups have a
local-to-global property for Morse quasi-geodesics. This allows us to
generalize combination theorems of Gitik for quasiconvex subgroups of
hyperbolic groups to the stable subgroups of these groups. In the case of the
mapping class group, this gives combination theorems for convex cocompact
subgroups. We show a number of additional consequences of this local-to-global
property, including a Cartan-Hadamard type theorem for detecting hyperbolicity
locally and discreteness of translation length of conjugacy classes of Morse
elements with a fixed gauge. To prove the relatively hyperbolic case, we
develop a theory of deep points for local quasi-geodesics in relatively
hyperbolic spaces, extending work of Hruska.
