Counting subgroups via Mirzakhani's curve counting

TL;DR

This work extends Mirzakhani’s curve counting to conjugacy classes of finitely generated subgroups of $\pi_1(\Sigma)$ on finite-area hyperbolic surfaces by introducing a generalized length $\ell_{\mathrm{SC}}$ defined as half the boundary length of the convex core and a boundary projection $\mathcal{B}$ on subset currents. The authors establish $L$-asymptotics with exponent $6g-6+2r$ and a positive constant $\mathfrak{c}_{g,r}^{\Gamma}(H)$ for each orbit under a finite-index subgroup $\Gamma\le \mathrm{Map}(\Sigma)$, with $\mathfrak{c}_{g,r}^{\Gamma}(H)>0$ unless $H$ is finite-index; cyclic and non-cyclic cases correspond to closed geodesics and subsurfaces, respectively. They reinterpret conjugacy classes as subsurfaces via convex cores and embed the counting problem into the broader framework of subset currents, using ES22 to support generalizations to arbitrary positive homogeneous functionals on currents. The paper also introduces an area functional on subset currents, proves convergence of counting measures to the Thurston measure, and derives corollaries for weighted sums and general functionals, thereby unifying curve counting and subsurface counting within a Currents-based paradigm and highlighting a natural completion of counting problems for subgroups.

Abstract

Given a hyperbolic surface $Σ$ 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 $π_1(Σ)$, we extend this to the counting problem of conjugacy classes of finitely generated subgroups of $π_1(Σ)$. 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 $Σ$ 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 $π_1(Σ)$.

Figures (5)

• Figure 1: The convex core $C_{\langle x,y\rangle}$ is described as the subsurface of $\Sigma$.
• Figure 2: The left of the figure is the covering graph $\Delta_H$ and the right of the figure is the covering graph $\Delta_{\phi(H)}$.
• Figure 3: This figure illustrates the convergence of the limit in Theorem \ref{['thm:ES']}.
• Figure 4: This figure illustrates the convergence of the limit in Theorem \ref{['thm:general counting thm']}. The key observation is that the area functional $\mathrm{Area} \colon \mathrm{SC}_K(\Sigma) \to \mathbb {R}_{\geq 0}$ is a continuous $\mathbb {R}_{\geq 0}$-linear $\mathrm{Map}$-invariant functional, and the function $\sqrt{i(\mathcal{B} (\cdot ),\mathcal{B} (\cdot ))}\colon \mathrm{SC}_K (\Sigma) \times \mathrm{SC}_K (\Sigma) \to \mathbb {R}_{\geq 0}$ is a continuous $\mathbb {R}_{\geq 0}$-linear $\mathrm{Map}$-invariant functional.
• Figure 5: In the left of the figure, $p_H(C_H)$ and $p_K(C_K)$ are described as subsurfaces of the closed surface $\Sigma$ of genus $8$. Their intersection $p_H(C_H)\cap p_K (C_K)$, which is blacked out, is a surface of genus $1$ with one boundary component.

Theorems & Definitions (35)

• Theorem 1: See Theorem \ref{['thm:counting subgroup use length']}
• Theorem 2: See Theorem \ref{['thm:general counting thm']}
• Corollary 3: See Corollary \ref{['cor:counting eta']}
• Definition 2.1: Subset current
• Definition 2.2: Geodesic current and Measured lamination
• Proposition 2.3: See Sas22
• Example 2.4
• Proposition 2.5
• proof
• Definition 2.6: Thurston measure