Matheus, Carlos; Yoccoz, Jean-Christophe; Zmiaikou, David
Homology of origamis with symmetries  [ Homologie des origamis avec symétries ]
Annales de l'institut Fourier, Tome 64 (2014) no. 3 , p. 1131-1176
MR 3330166 | Zbl 06387303
doi : 10.5802/aif.2876
URL stable : http://www.numdam.org/item?id=AIF_2014__64_3_1131_0

Classification:  37D40,  30F10,  32G15,  20C05
Mots 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
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.

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