Abstract
Given a connected, oriented, complete, finite-area hyperbolic surface of genus with punctures, Mirzakhani showed that the number of simple closed multigeodesics on of a prescribed topological type and total hyperbolic length is asymptotic to a polynomial in of degree as . 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.
Citation
Francisco Arana-Herrera. "Counting hyperbolic multigeodesics with respect to the lengths of individual components and asymptotics of Weil–Petersson volumes." Geom. Topol. 26 (3) 1291 - 1347, 2022. https://doi.org/10.2140/gt.2022.26.1291
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