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Institut de Mathématiques de Marseille, UMR 7373
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Rauzy-Veech groups of flat surfaces

11h00 - 12h00
horaire FRUMAM

Aix-Marseille Université - Site St Charles
3, place Victor Hugo - case 39
13331 MARSEILLE Cedex 03

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. Moreover, some techniques can be extended to the quadratic case to prove that the indices of the Rauzy–Veech groups of certain connected components of quadratic strata are also finite. This proves that the “plus” Lyapunov spectra of such strata is simple, which was not previously known.