# Mirzakhani's Curve Counting: From Simple to All

## TL;DR

This paper presents a straightforward method to extend Mirzakhani's curve counting results from simple curves to all curves, simplifying the understanding of asymptotic growth in hyperbolic geometry.

## Contribution

It provides a low-tech, accessible derivation of the general curve counting result from the simple curve case.

## Key findings

- Extended Mirzakhani's asymptotic growth results to all curves
- Provided a simplified derivation method
- Enhanced understanding of curve counting in hyperbolic surfaces

## Abstract

Mirzakhani obtained the asymptotic growth, when $L\to\infty$, of the number of curves in the mapping class group orbit of some given simple curve and with length at most $L$. Years later she extended this result from simple to arbitrary curves. Here we give a short and relative low-tech argument showing how to derive the general result from the one for simple curves.