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

TL;DR

This paper explicitly computes how affine diffeomorphisms act on the relative homology of two highly symmetric origamis, revealing finite group actions and invariant root systems, and providing new proofs for known Lyapunov exponent results.

Contribution

It provides explicit calculations of the affine diffeomorphism actions on homology for specific symmetric origamis, linking these actions to root systems and Lyapunov exponents.

Findings

Action on homology is through finite groups

Invariant subspaces contain a D4 root system

Lyapunov exponents are zero for these origamis

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-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.