Growth rates, stable subgroups, and regular languages

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 $A$ 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 $\mathrm{Out}(F_n)$, 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.