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

Francisco Arana-Herrera

Geometry & Topology 26 (2022) 1291–1347

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 L is asymptotic to a polynomial in L of degree 6g 6 + 2n as L . 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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counting, hyperbolic, multigeodesics, Weil-Petersson, volumes
Mathematical Subject Classification
Primary: 30F60
Secondary: 32G15
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
Francisco Arana-Herrera
Department of Mathematics
Stanford University
Stanford, CA
United States