Counting problems from the viewpoint of ergodic theory: from primitive integer points to simple closed curves

TL;DR

This paper surveys Mirzakhani's proof that the number of simple closed geodesics of length ≤ L on a hyperbolic surface grows asymptotically as a constant times L^{6g-6}, connecting primitive lattice point counting and hyperbolic geometry.

Contribution

It provides a detailed, accessible account of Mirzakhani's proof, highlighting the connection between primitive lattice counting and geodesic enumeration on hyperbolic surfaces.

Findings

Asymptotic growth rate of simple closed geodesics is L^{6g-6}

Connection between lattice point counting and hyperbolic geometry principles

Detailed explanation suitable for non-experts

Abstract

In her thesis, Mirzakhani showed that the number of simple closed geodesics of length $\leq L$ on a closed, connected, oriented hyperbolic surface $X$ of genus $g$ is asymptotic to $L^{6g-6}$ times a constant depending on the geometry of $X$. In this survey we give a detailed account of Mirzakhani's proof of this result aimed at non-experts. We draw inspiration from classic primitive lattice point counting results in homogeneous dynamics. The focus is on understanding how the general principles that drive the proof in the case of lattices also apply in the setting of hyperbolic surfaces.