Today Jean-Christophe Yoccoz and I uploaded to the arXiv our joint paper “The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis“. Since the goal of this paper was already explained in this previous post (where the reader can also find some slides of a talk I gave at Orsay a few months ago), here I will only reproduce the abstract of the 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 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.