A combinatorial approach to Rauzy-type dynamics I: permutations and the Kontsevich--Zorich--Boissy classification theorem

Abstract

Rauzy-type dynamics are group actions on a collection of combinatorial objects. The first and best known example concerns an action on permutations, associated to interval exchange transformations (IET) for the Poincar\'e map on compact orientable translation surfaces. The equivalence classes on the objects induced by the group action are related to components of the moduli spaces of Abelian differentials with prescribed singularities, and, in two variants of the problem, have been classified by Kontsevich and Zorich, and by Boissy, through methods involving both combinatorics and algebraic geometry. We provide here a purely combinatorial proof of both classification theorems, and in passing establish a few previously unnoticed features. As will be shown elsewhere, our methods extend also to other Rauzy-type dynamics, both on labeled and unlabeled structures. Some of these dynamics have a geometrical interpretation (e.g., matchings, related to IET on non-orientable surfaces), while some others do not have one so far.