The Rauzy–Veech induction is a powerful renormalization procedure for (half-)translation flows. By tracking the changes it induces in homology, we define the Rauzy–Veech monoid (or group) of a connected component of a stratum of Abelian or quadratic differentials. This monoid was proven to be pinching and twisting by Avila and Viana, which implies the Kontsevich–Zorich conjecture stating the simplicity of the Lyapunov spectrum of almost every translation flow with respect to the Masur–Veech measures.

 

In this talk, I will present a full classification of the Rauzy–Veech groups of Abelian strata: they are explicit finite-index subgroups of their ambient symplectic groups. This is strictly stronger than pinching and twisting and solves a conjecture of Zorich about the Zariski-density of such groups. Moreover, some techniques can be extended to the quadratic case to prove that the indices of the “plus” and/or “minus” Rauzy–Veech groups of certain connected components of quadratic strata are also finite. This proves that the Lyapunov spectra of such strata are simple, which was not previously known.

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