Counting lattice points in moduli spaces of quadratic differentials
Vincent Delecroix
Laboratoire Bordelais de Recherche en Informatique, Université de Bordeaux, Talence, FranceÉlise Goujard
Institut de Mathématiques de Bordeaux, Université de Bordeaux, Talence, FrancePeter Zograf
Euler International Mathematical Institute, St. Petersburg, RussiaAnton Zorich
Center for Advanced Studies, Skoltech, Moscow, Russia, and Institut de Mathématiques de Jussieu–Paris Rive Gauche, Paris, France
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Abstract
We show how to count lattice points represented by square-tiled surfaces in the moduli spaces of meromorphic quadratic differentials with simple poles on complex algebraic curves. We demonstrate the versatility of the lattice point count on three different examples, including evaluation of Masur–Veech volumes of the moduli spaces of quadratic differentials, computation of asymptotic frequencies of geodesic multicurves on hyperbolic surfaces, and asymptotic enumeration of meanders with a fixed number of minimal arcs.