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

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

2009-12-08

arxiv.org/abs/0912.1425v2

Journal of Modern Dynamics, vol. 4, n. 3, 453--486 (2010)

Mathematics

Dynamical Systems

Final revision (based on the referees reports). 33 pages, 10 figures

Scientific paper

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-Matheus (in genus 4). We show that, in both cases, the action on the non trivial 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 non trivial Lyapunov exponents of the Kontsevich-Zorich cocycle for the Teichmuller disks of these two origamis are equal to zero.

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