2010, 4(3): 453-486. doi: 10.3934/jmd.2010.4.453

The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis

1. 

Collège de France, 3 Rue d’Ulm, 75005, Paris, France, France

Received  December 2009 Revised  June 2010 Published  October 2010

We compute explicitly the action of the group of affine diffeomorphisms on the relative homology of two remarkable origamis discovered respectively by Forni (in genus $3$) and Forni and Matheus (in genus $4$). We show that, in both cases, the action on the nontrivial part of the homology is through finite groups. In particular, the action on some $4$-dimensional invariant subspace of the homology leaves invariant a root system of $D_4$ type. This provides as a by-product a new proof of (slightly stronger versions of) the results of Forni and Matheus: the nontrivial Lyapunov exponents of the Kontsevich-Zorich cocycle for the Teichmüller disks of these two origamis are equal to zero.
Citation: Carlos Matheus, Jean-Christophe Yoccoz. The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis. Journal of Modern Dynamics, 2010, 4 (3) : 453-486. doi: 10.3934/jmd.2010.4.453
References:
[1]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture,, Acta Math., 198 (2007), 1. doi: doi:10.1007/s11511-007-0012-1.

[2]

O. Bauer, Familien von Jacobivarietäten über Origamikurven,, PhD thesis, (2009).

[3]

J. Borwein and P. Borwein, "Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity,", Canadian Math. Soc. Series of Monographs and Advanced Texts, (1987).

[4]

N. Bourbaki, "Groupes et Algèbres de Lie. Chapitre VI: Systèmes de Racines,", Hermann, (1960).

[5]

I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents,, to appear in Annals of Math., ().

[6]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math., 155 (2002), 1. doi: doi:10.2307/3062150.

[7]

G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, Handbook of Dynamical Systems, 1B (2006), 549.

[8]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum,, preprint, ().

[9]

F. Herrlich and G. Schmithüsen, An extraordinary origami curve,, Math. Nachr., 281 (2008), 219. doi: doi:10.1002/mana.200510597.

[10]

P. Hubert and T. Schmidt, An introduction to Veech surfaces,, Handbook of Dynamical Systems, 1B (2006), 501.

[11]

M. Kontsevich, Lyapunov exponents and Hodge theory,, in, 24 (1996), 318.

[12]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,, Invent. Math., 153 (2003), 631. doi: doi:10.1007/s00222-003-0303-x.

[13]

E. Lanneau, Connected components of the strata of the moduli spaces of quadratic differentials,, Ann. Sci. ENS, 41 (2008), 1.

[14]

H. Masur, Interval-exchange transformations and measured foliations,, Ann. of Math., 115 (1982), 169. doi: doi:10.2307/1971341.

[15]

M. Möller, Shimura and Teichmüller curves,, preprint, ().

[16]

W. Veech, Teichmüller geodesic flow,, Ann. of Math., 124 (1986), 441. doi: doi:10.2307/2007091.

[17]

W. Veech, Gauss measures for transformations on the space of interval-exchange maps,, Ann. of Math., 115 (1982), 201. doi: doi:10.2307/1971391.

[18]

W. Veech, Moduli spaces of quadratic differentials,, J. Anal. Math., 55 (1990), 117. doi: doi:10.1007/BF02789200.

[19]

W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards,, Inv. Math., 97 (1989), 553. doi: doi:10.1007/BF01388890.

[20]

J. C. Yoccoz, Interval-exchange maps and translation surfaces,, Clay Math. Inst. Summer School on Homogenous Flows, (2007).

[21]

A. Zorich, Asymptotic flag of an orientable measured foliation on a surface,, in, (1994), 479.

[22]

A. Zorich, Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials,, Journal of Modern Dynamics, 2 (2008), 139.

[23]

A. Zorich, Flat surfaces,, Frontiers in Number Theory, Physics, and Geometry, (2006), 437. doi: doi:10.1007/978-3-540-31347-2_13.

show all references

References:
[1]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture,, Acta Math., 198 (2007), 1. doi: doi:10.1007/s11511-007-0012-1.

[2]

O. Bauer, Familien von Jacobivarietäten über Origamikurven,, PhD thesis, (2009).

[3]

J. Borwein and P. Borwein, "Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity,", Canadian Math. Soc. Series of Monographs and Advanced Texts, (1987).

[4]

N. Bourbaki, "Groupes et Algèbres de Lie. Chapitre VI: Systèmes de Racines,", Hermann, (1960).

[5]

I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents,, to appear in Annals of Math., ().

[6]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math., 155 (2002), 1. doi: doi:10.2307/3062150.

[7]

G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, Handbook of Dynamical Systems, 1B (2006), 549.

[8]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum,, preprint, ().

[9]

F. Herrlich and G. Schmithüsen, An extraordinary origami curve,, Math. Nachr., 281 (2008), 219. doi: doi:10.1002/mana.200510597.

[10]

P. Hubert and T. Schmidt, An introduction to Veech surfaces,, Handbook of Dynamical Systems, 1B (2006), 501.

[11]

M. Kontsevich, Lyapunov exponents and Hodge theory,, in, 24 (1996), 318.

[12]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,, Invent. Math., 153 (2003), 631. doi: doi:10.1007/s00222-003-0303-x.

[13]

E. Lanneau, Connected components of the strata of the moduli spaces of quadratic differentials,, Ann. Sci. ENS, 41 (2008), 1.

[14]

H. Masur, Interval-exchange transformations and measured foliations,, Ann. of Math., 115 (1982), 169. doi: doi:10.2307/1971341.

[15]

M. Möller, Shimura and Teichmüller curves,, preprint, ().

[16]

W. Veech, Teichmüller geodesic flow,, Ann. of Math., 124 (1986), 441. doi: doi:10.2307/2007091.

[17]

W. Veech, Gauss measures for transformations on the space of interval-exchange maps,, Ann. of Math., 115 (1982), 201. doi: doi:10.2307/1971391.

[18]

W. Veech, Moduli spaces of quadratic differentials,, J. Anal. Math., 55 (1990), 117. doi: doi:10.1007/BF02789200.

[19]

W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards,, Inv. Math., 97 (1989), 553. doi: doi:10.1007/BF01388890.

[20]

J. C. Yoccoz, Interval-exchange maps and translation surfaces,, Clay Math. Inst. Summer School on Homogenous Flows, (2007).

[21]

A. Zorich, Asymptotic flag of an orientable measured foliation on a surface,, in, (1994), 479.

[22]

A. Zorich, Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials,, Journal of Modern Dynamics, 2 (2008), 139.

[23]

A. Zorich, Flat surfaces,, Frontiers in Number Theory, Physics, and Geometry, (2006), 437. doi: doi:10.1007/978-3-540-31347-2_13.

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