  We prove that Morse local-to-global groups grow exponentially faster than
their infinite index stable subgroups. This generalizes a result of Dahmani,
Futer, and Wise in the context of quasi-convex subgroups of hyperbolic groups
to a broad class of groups that contains the mapping class group, CAT(0)
groups, and the fundamental groups of closed 3-manifolds. To accomplish this,
we develop a theory of automatic structures on Morse geodesics in Morse
local-to-global groups. Other applications of these automatic structures
include a description of stable subgroups in terms of regular languages,
rationality of the growth of stable subgroups, density in the Morse boundary of
the attracting fixed points of Morse elements, and containment of the Morse
boundary inside the limit set of any infinite normal subgroup.
