Homology of origamis with symmetries
[Homologie des origamis avec symétries]
Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 1131-1176.

Étant donné un origami (surface à petits carreaux) M avec un groupe d’automorphismes Γ, nous déterminons la décomposition du premier groupe d’homologie de M en Γ-submodules isotypiques. Parmi l’action du groupe affine de M sur le groupe d’homologie, nous déduisons quelques conséquences pour les multiplicités des exposants de Lyapunov du cocycle de Kontsevich-Zorich. De plus, nous construisons et étudions plusieurs familles d’origamis intéressants pour illustrer nos résultats.

Given an origami (square-tiled surface) M with automorphism group Γ, we compute the decomposition of the first homology group of M into isotypic Γ-submodules. Through the action of the affine group of M on the homology group, we deduce some consequences for the multiplicities of the Lyapunov exponents of the Kontsevich-Zorich cocycle. We also construct and study several families of interesting origamis illustrating our results.

DOI : 10.5802/aif.2876
Classification : 37D40, 30F10, 32G15, 20C05
Keywords: Origamis, square-tiled surfaces, automorphisms group, affine group, representations of finite groups, regular and quasi-regular origamis, Kontsevich-Zorich cocycle, Lyapunov exponents
Mot clés : origamis, surfaces à petits carreaux, groupes d’automorphismes, groupes affines, représentations des groupes finis, origamis réguliers et quasi-réguliers, cocycle de Kontsevich-Zorich, exposants de Lyapunov
Matheus, Carlos 1 ; Yoccoz, Jean-Christophe 2 ; Zmiaikou, David 3

1 Université Paris 13 Sorbonne Paris Cité LAGA, CNRS (UMR 7539) F-93430, Villetaneuse (France)
2 Collège de France (PSL) 3, Rue d’Ulm 75005 Paris (France)
3 Département de Mathématiques Université Paris-Sud 11 91405 Orsay Cedex (France)
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Matheus, Carlos; Yoccoz, Jean-Christophe; Zmiaikou, David. Homology of origamis with symmetries. Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 1131-1176. doi : 10.5802/aif.2876. http://www.numdam.org/articles/10.5802/aif.2876/

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