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

TL;DR

This paper offers a purely combinatorial proof of the Kontsevich--Zorich and Boissy classification theorems for Rauzy-type dynamics on permutations, revealing new features and extending to other dynamics.

Contribution

It provides a novel combinatorial proof of key classification theorems and introduces methods applicable to various Rauzy-type dynamics beyond the original scope.

Findings

Purely combinatorial proof of classification theorems

Identification of previously unnoticed features

Extension of methods to other Rauzy-type dynamics

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.