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

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 $\leq L$ is asymptotic to a polynomial in $L$ of degree $6 g - 6 + 2 n$ 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.

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Keywords

counting, hyperbolic, multigeodesics, Weil-Petersson, volumes

Mathematical Subject Classification

Primary: 30F60

Secondary: 32G15

References

Publication

Received: 14 April 2020

Revised: 7 December 2020

Accepted: 14 January 2021

Published: 3 August 2022

Proposed: David M Fisher

Seconded: Mladen Bestvina, Benson Farb

Authors

Francisco Arana-Herrera
Department of Mathematics Stanford University Stanford, CA United States

Volume 26, issue 3 (2022)

Francisco Arana-Herrera

Abstract