Counting hyperbolic multigeodesics with respect to the lengths of individual components and asymptotics of Weil–Petersson volumes

Abstract

Given a connected, oriented, complete, finite-area hyperbolic surface $X$ of genus $g$ with $n$ punctures, Mirzakhani showed that the number of simple closed multigeodesics on $X$ of a prescribed topological type and total hyperbolic length $\le L$ is asymptotic to a polynomial in $L$ of degree $6g-6+2n$ as $L\to \infty$. We establish asymptotics of the same kind for counts of simple closed multigeodesics that keep track of the hyperbolic length of individual components rather than just the total hyperbolic length, proving a conjecture of Wolpert. The leading terms of these asymptotics are related to limits of Weil–Petersson volumes of expanding subsets of quotients of Teichmüller space. We introduce a framework for computing limits of this kind in terms of purely topological information. We provide two further applications of this framework to counts of square-tiled surfaces and counts of filling closed multigeodesics on hyperbolic surfaces.