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We study relations between Rauzy classes coming from an interval exchange mapand the corresponding connected components of strata of the moduli space ofAbelian differentials. This gives a criterion to decide whether twopermutations are in the same Rauzy class or not, without actually computingthem. We prove a similar result for Rauzy classes corresponding to quadraticdifferentials.

Classification of Rauzy classes in the moduli space of quadratic differentials