Counting subgroups via Mirzakhani's curve counting

Abstract

Given a hyperbolic surface $\Sigma$ of genus $g$ with $r$ cusps, Mirzakhani proved that the number of closed geodesics of length at most $L$ and of a given type is asymptotic to $cL^{6g-6+2r}$ for some $c>0$. Since a closed geodesic corresponds to a conjugacy class of the fundamental group $\pi_1(\Sigma )$, we extend this to the counting problem of conjugacy classes of finitely generated subgroups of $\pi_1(\Sigma )$. Using `half the sum of the lengths of the boundaries of the convex core of a subgroup' instead of the length of a closed geodesic, we prove that the number of such conjugacy classes is similarly asymptotic to $cL^{6g-6+2r}$ for some $c>0$. Furthermore, we see that this measurement for subgroups is `natural' within the framework of subset currents, which serve as a completion of weighted conjugacy classes of finitely generated subgroups of $\pi_1(\Sigma )$.