Counting subgroups via Mirzakhani's curve counting

TL;DR

This paper extends Mirzakhani's geodesic counting results to finitely generated subgroups of surface groups, establishing asymptotic counts for conjugacy classes and subsurfaces using a natural length measure within subset currents.

Contribution

It introduces a new counting method for subgroups of surface groups, generalizing Mirzakhani's geodesic counting theorem to a broader setting.

Findings

Asymptotic count of conjugacy classes of subgroups matches Mirzakhani's geodesic count

Uses a natural length measure related to convex cores of subgroups

Connects subgroup counting with subset currents framework

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 $c>0$. 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 $\pi_1(\Sigma )$.