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

We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich–Zorich cocycle over the Teichmüller flow on the L2(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 ⩾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.