We show that the language of geodesic words representing elements of a stable subgroup H of a group G with finite generating set A is regular, and that there is a sublanguage which bijects H. Consequently, the growth function of H with respect to A
Nearby in the stack
is rational, and in many cases, one can deduce a growth rate gap between
H
and
G
. In particular, this applies to convex cocompact subgroups of
Out(Fn)
, handlebody groups, and Torelli groups of surfaces of sufficient complexity. We also provide an example of a finitely presented, relatively hyperbolic, and Morse local-to-global group which contains a stable subgroup with unsolvable membership problem, answering a question of Cordes, Russell, Spriano, and Zalloum.