Homology of origamis with symmetries

Abstract

Given an origami (square-tiled surface) M with automorphism group G, we compute the decomposition of the first homology group of M into isotypic G-submodules. Through the action of the affine group of M on the homology group, we deduce some consequences for the multiplicities of the Lyapunov exponents of the Kontsevich-Zorich cocycle. We also construct and study several families of interesting origamis illustrating our results.

36 pages, no figures. Final version incorporating the referee's comments. To appear in Annales de l'Institut Fourier, Volume 64 (2014)