A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces

Abstract

We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle over the Teichmueller flow on the $S{L}_2(R)$-orbit of a square-tiled surface. The simplicity of the Lyapunov spectrum has been proved by A. Avila and M. Viana with respect to the so-called Masur-Veech measures associated to connected components of moduli spaces of translation surfaces, but is not always true for square-tiled surfaces of genus $\ge 3$. We apply our criterion to square-tiled surfaces of genus 3 with one single zero. Conditionally to a conjecture of Delecroix and Lelièvre, we prove with the aid of Siegel's theorem (on integral points on algebraic curves of genus $>0$) that all but finitely many such square-tiled surfaces have simple Lyapunov spectrum.

69 pages, Appendix C by Samuel Lelièvre. Final version based on the referees' reports. To appear in Invent. Math

References in corpus (9)

• Connected components of the moduli spaces of Abelian differentials with prescribed singularities
• Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow
• Invariant and stationary measures for the SL(2,R) action on Moduli space
• Lyapunov exponents and Hodge theory
• Euler characteristics of Teichmüller curves in genus two
• Square-tiled cyclic covers
• Complete periodicity of Prym eigenforms
• Teichmueller curves generated by Weierstrass Prym eigenforms in genus three and genus four
• Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces

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• Convergence of Siegel-Veech constants
• Diagonal flow detects topology of strata
• Quelques contributions à la théorie de l'action de SL(2,R) sur les espaces de modules de surfaces plates