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 $c{L}^{6g-6+2r}$ 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 $c{L}^{6g-6+2r}$ for some $c>0$. As a special case, these conjugacy classes can be interpreted as subsurfaces of $\Sigma$ via their convex cores, and the result can be viewed as counting subsurfaces of a given type. Furthermore, we see that the above length 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 ${\pi }_1(\Sigma )$.