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We prove that Morse local-to-global groups grow exponentially faster thantheir infinite index stable subgroups. This generalizes a result of Dahmani,Futer, and Wise in the context of quasi-convex subgroups of hyperbolic groupsto 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 Morselocal-to-global groups. Other applications of these automatic structuresinclude a description of stable subgroups in terms of regular languages,rationality of the growth of stable subgroups, density in the Morse boundary ofthe attracting fixed points of Morse elements, and containment of the Morseboundary inside the limit set of any infinite normal subgroup.

Regularity of Morse geodesics and growth of stable subgroups