Counting hyperbolic multi-geodesics with respect to the lengths of individual components

TL;DR

This paper extends Mirzakhani's asymptotic counting of multi-geodesics on hyperbolic surfaces to include the lengths of individual components, confirming a conjecture of Wolpert and providing detailed geometric and topological descriptions.

Contribution

It establishes new asymptotic formulas for counting multi-geodesics with respect to individual component lengths, generalizing previous results and confirming Wolpert's conjecture.

Findings

Asymptotics for multi-geodesics with component length tracking

Generalization of Mirzakhani's counting results

Confirmation of Wolpert's conjecture

Abstract

Given a connected, oriented, complete, finite area hyperbolic surface $X$ of genus $g$ with $n$ punctures, Mirzakhani showed that the number of multi-geodesics on $X$ of total hyperbolic length $\leq L$ in the mapping class group orbit of a given simple or filling closed multi-curve is asymptotic as $L \to \infty$ to a polynomial in $L$ of degree $6g-6+2n$. We establish asymptotics of the same kind for countings of multi-geodesics in mapping class group orbits of simple or filling closed multi-curves that keep track of the hyperbolic lengths of individual components, proving and generalizing a conjecture of Wolpert. In the simple case we consider more precise countings that also keep track of the class of the multi-geodesics in the space of projective measured geodesic laminations. We provide a unified geometric and topological description of the leading terms of the asymptotics of all the countings considered. Our proofs combine techniques and results from several papers of Mirzakhani as well as ideas introduced by Margulis in his thesis.