Counting Mapping Class group orbits on hyperbolic surfaces

TL;DR

This paper investigates the asymptotic growth of lengths of closed curves of fixed topological type on hyperbolic surfaces, providing insights into curve counting, self-intersection growth, and properties of random pants decompositions using ergodic earthquake flow.

Contribution

It introduces new asymptotic formulas for counting closed curves and analyzes random pants decompositions on hyperbolic surfaces, leveraging ergodic properties of earthquake flow.

Findings

Asymptotic growth rates for lengths of fixed topological type curves.

Growth estimates for the number of curves with bounded length and self-intersections.

Properties of random pants decompositions at large lengths.

Abstract

Let $S_{g,n}$ be a surface of genus $g $ with $n$ marked points. Let $X$ be a complete hyperbolic metric on $S_{g,n}$ with $n$ cusps. Every isotopy class $[\gamma]$ of a closed curve $\gamma \in \pi_{1}(S_{g,n})$ contains a unique closed geodesic on $X$. Let $\ell_{\gamma}(X)$ denote the hyperbolic length of the geodesic representative of $\gamma$ on $X$. In this paper, we study the asymptotic growth of the lengths of closed curves of a fixed topological type on $S_{g,n}.$ As an application, one can obtain the asymptotics of the growth of $s^{k}_{X}(L)$, the number of closed curves of length $\leq L$ on $X$ with at most $k$ self-intersections. We also discuss properties of random pants decomposition of large length on $X$. Both these results are based on ergodic properties of the earthquake flow on a natural bundle over the moduli space $\mathcal{M}_{g,n}$ of hyperbolic surfaces of genus $g$ with $n$ cusps.