Homology of origamis with symmetries

Matheus, Carlos; Yoccoz, Jean-Christophe; Zmiaikou, David

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 $\Gamma$, nous déterminons la décomposition du premier groupe d’homologie de $M$ en $\Gamma$-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 $\Gamma$, we compute the decomposition of the first homology group of $M$ into isotypic $\Gamma$-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.

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

[1] Avila, Artur; Viana, Marcelo Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture, Acta Math., Volume 198 (2007) no. 1, pp. 1-56 | MR | Zbl

[2] Bainbridge, Matt Euler characteristics of Teichmüller curves in genus two, Geom. Topol., Volume 11 (2007), pp. 1887-2073 | MR | Zbl

[3] Bouw, Irene I.; Möller, Martin Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2), Volume 172 (2010) no. 1, pp. 139-185 | MR | Zbl

[4] Chen, Dawei; Möller, Martin Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus, Geom. Topol., Volume 16 (2012) no. 4, pp. 2427-2479 | MR | Zbl

[5] Cornulier, Y. Formes bilinéaires invariantes (2004) (at http://www.normalesup.org/~cornulier/bil_inv.pdf)

[6] Delecroix, V.; Hubert, P.; Lelièvre, S. Diffusion for the periodic wind-tree model (arXiv:1107.1810)

[7] Eskin, Alex; Kontsevich, Maxim; Zorich, Anton Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow (Prprint arXiv:1112.5872)

[8] Eskin, Alex; Kontsevich, Maxim; Zorich, Anton Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., Volume 5 (2011) no. 2, pp. 319-353 | MR | Zbl

[9] Forni, Giovanni Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), Volume 155 (2002) no. 1, pp. 1-103 | MR | Zbl

[10] Fulton, William; Harris, Joe Representation theory, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991, pp. xvi+551 (A first course, Readings in Mathematics) | MR | Zbl

[11] Gutkin, Eugene; Judge, Chris Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J., Volume 103 (2000) no. 2, pp. 191-213 | MR | Zbl

[12] Herrlich, Frank Teichmüller curves defined by characteristic origamis, The geometry of Riemann surfaces and abelian varieties (Contemp. Math.), Volume 397, Amer. Math. Soc., Providence, RI, 2006, pp. 133-144 | MR | Zbl

[13] Herrlich, Frank; Schmithüsen, Gabriela An extraordinary origami curve, Math. Nachr., Volume 281 (2008) no. 2, pp. 219-237 | MR | Zbl

[14] Kontsevich, M. Lyapunov exponents and Hodge theory, The mathematical beauty of physics (Saclay, 1996) (Adv. Ser. Math. Phys.), Volume 24, World Sci. Publ., River Edge, NJ, 1997, pp. 318-332 | MR | Zbl

[15] Masur, Howard Interval exchange transformations and measured foliations, Ann. of Math. (2), Volume 115 (1982) no. 1, pp. 169-200 | MR | Zbl

[16] Matheus, Carlos; Yoccoz, Jean-Christophe The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, J. Mod. Dyn., Volume 4 (2010) no. 3, pp. 453-486 | MR | Zbl

[17] Ore, Oystein Some remarks on commutators, Proc. Amer. Math. Soc., Volume 2 (1951), pp. 307-314 | MR | Zbl

[18] Serre, Jean-Pierre Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977, pp. x+170 (Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42) | MR | Zbl

[19] Veech, William A. Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), Volume 115 (1982) no. 1, pp. 201-242 | MR | Zbl

[20] Veech, William A. The Teichmüller geodesic flow, Ann. of Math. (2), Volume 124 (1986) no. 3, pp. 441-530 | MR | Zbl

[21] Wilkinson, Amie Conservative partially hyperbolic dynamics, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi (2010), pp. 1816-1836 | MR | Zbl

[22] Yoccoz, Jean-Christophe Interval exchange maps and translation surfaces, Homogeneous flows, moduli spaces and arithmetic (Clay Math. Proc.), Volume 10, Amer. Math. Soc., Providence, RI, 2010, pp. 1-69 (available at http://www.college-de-france.fr/media/equ_dif/UPL15305_PisaLecturesJCY2007.pdf) | MR | Zbl

[23] Yu, Fei; Zuo, Kang Weierstrass filtration on Teichmüller curves and Lyapunov exponents, J. Mod. Dyn., Volume 7 (2013) no. 2, pp. 209-237 | MR | Zbl

[24] Zmiaikou, D. Origamis and permutation groups (2011) (PhD thesis, at http://www.zmiaikou.com/research)

[25] Zorich, Anton Asymptotic flag of an orientable measured foliation on a surface, Geometric study of foliations (Tokyo, 1993), World Sci. Publ., River Edge, NJ, 1994, pp. 479-498 | Numdam | MR | Zbl

[26] Zorich, Anton Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), Volume 46 (1996) no. 2, pp. 325-370 | Numdam | MR | Zbl

[27] Zorich, Anton Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, Volume 17 (1997) no. 6, pp. 1477-1499 | MR | Zbl

[28] Zorich, Anton On hyperplane sections of periodic surfaces, Solitons, geometry, and topology: on the crossroad (Amer. Math. Soc. Transl. Ser. 2), Volume 179, Amer. Math. Soc., Providence, RI, 1997, pp. 173-189 | MR | Zbl

[29] Zorich, Anton How do the leaves of a closed $1$-form wind around a surface?, Pseudoperiodic topology (Amer. Math. Soc. Transl. Ser. 2), Volume 197, Amer. Math. Soc., Providence, RI, 1999, pp. 135-178 | MR | Zbl

[30] Zorich, Anton Flat surfaces, Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, pp. 437-583 | MR | Zbl