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

Carlos Matheus; Jean-Christophe Yoccoz

doi:10.3934/jmd.2010.4.453

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2010, Volume 4, Issue 3: 453-486. Doi: 10.3934/jmd.2010.4.453

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

Carlos Matheus^1, and
Jean-Christophe Yoccoz^1,

1.
Collège de France, 3 Rue d’Ulm, 75005, Paris

Received: December 2009

Revised: June 2010

Published: September 2010

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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.

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Mathematics Subject Classification: Primary: 37D40; Secondary: 37Axx.

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