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Journal of Modern Dynamics (JMD)
 

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

Pages: 453 - 486, Issue 3, July 2010      doi:10.3934/jmd.2010.4.453

 
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Carlos Matheus - Collège de France, 3 Rue d’Ulm, 75005, Paris, France (email)
Jean-Christophe Yoccoz - Collège de France, 3 Rue d’Ulm, 75005, Paris, France (email)

Abstract: 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.

Keywords:  Teichmüller dynamics, Kontsevich-Zorich cocycle, totally degenerate origamis.
Mathematics Subject Classification:  Primary: 37D40; Secondary: 37Axx.

Received: December 2009;      Revised: June 2010;      Available Online: October 2010.

 References