Higher genus meanders and Masur–Veech volumes
Delecroix, Vincent1; Goujard, Élise2; Zograf, Peter3; Zorich, Anton4
1LaBRI, Domaine universitaire, 351 cours de la Libération, 33405 Talence (France)
2IMB, Univ. de Bordeaux, 351 cours de la Libération, 33405 Talence (France), and, Institut Universitaire de France
3St. Petersburg Department, Steklov Math. Institute, Fontanka 27, St. Petersburg 191023 (Russia), and, Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O. 29B, St.Petersburg 199178 (Russia)
4Institut de Mathématiques de Jussieu –, Paris Rive Gauche, Case 7012, 8 Place Aurélie Nemours, 75205 Paris Cedex 13 (France)
Annales de l'Institut Fourier, Online first, 73 p.

A meander is a pair consisting of a straight line in the plane and of a smooth closed curve transversally intersecting the line, considered up to an isotopy preserving the straight line. The number of meanders with $2N$ intersections grows exponentially with $N$, but asymptotics still remains conjectural.

A meander defines a pair of transversally intersecting simple closed curves on a $2$-sphere. In this paper we consider such pairs on a closed oriented surface of arbitrary genus. The number of these higher genus meanders still admits exponential upper and lower bounds as $N$ grows. Fixing the number $n$ of bigons in the complement to the union of the two curves, we compute the precise asymptotics of genus $g$ meanders with $n$ bigons and with at most $2N$ intersections and show that it grows polynomially with $N$. We obtain a similar result in the case of oriented curves.

Un méandre est une paire formée d’une ligne droite dans le plan et d’une courbe fermée lisse intersectant la ligne transversalement, considérée à isotopie préservant la ligne près. Le nombre de méandres à $2N$ intersections croît exponentiellement en $N$, mais son asymptotique est encore conjecturale.

Un méandre définit une paire de courbes fermées simples s’intersectant transversalement sur la sphère. Dans cet article nous considérons de telles paires sur des surfaces orientées fermées de genre arbitraire. Le nombre de tels méandres admet également des bornes supérieure et inférieure exponentielles en $N$. En fixant le nombre $n$ de bigones dans le complément de l’union des deux courbes, nous calculons l’asymptotique précise du nombre de méandres de genre $g$ à $n$ bigones et au plus $2N$ intersections et montrons qu’elle est polynomiale en $N$. Nous obtenons des résultats similaires dans le cas de courbes orientées.

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DOI: 10.5802/aif.3759
Classification: 05A16, 32G15, 30F60, 60B05, 51H20
Keywords: meanders, curves on surfaces, moduli spaces, Masur–Veech volumes
Mots-clés : méandres, courbes sur les surfaces, espaces de modules, volumes de Masur–Veech

Delecroix, Vincent  1 ; Goujard, Élise  2 ; Zograf, Peter  3 ; Zorich, Anton  4

1 LaBRI, Domaine universitaire, 351 cours de la Libération, 33405 Talence (France)
2 IMB, Univ. de Bordeaux, 351 cours de la Libération, 33405 Talence (France), and, Institut Universitaire de France
3 St. Petersburg Department, Steklov Math. Institute, Fontanka 27, St. Petersburg 191023 (Russia), and, Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O. 29B, St.Petersburg 199178 (Russia)
4 Institut de Mathématiques de Jussieu –, Paris Rive Gauche, Case 7012, 8 Place Aurélie Nemours, 75205 Paris Cedex 13 (France) 
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Delecroix, Vincent; Goujard, Élise; Zograf, Peter; Zorich, Anton. Higher genus meanders and Masur–Veech volumes. Annales de l'Institut Fourier, Online first, 73 p.

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