Simplicity of the Lyapunov spectra of certain quadratic differentials

Abstract

We prove that the "plus" Rauzy--Veech groups of the connected components of all strata of meromorphic quadratic differentials defined on Riemann surfaces of genus at least one having at most simple poles and at least three singularities (zeros or poles), not all of even order, are finite-index subgroups of their ambient symplectic groups. This shows that the "plus" Lyapunov spectrum of such strata is simple. Moreover, we show that the index of the "minus" Rauzy--Veech group is also finite for connected components of strata satisfying the same conditions and having exactly two singularities of odd order. This shows that the "minus" Lyapunov spectrum of such strata is simple.