A Combinatorial Proof of the Kontsevich-Zorich-Boissy Classification of Rauzy Classes

TL;DR

This paper provides a purely combinatorial proof of the classification of Rauzy and Extended Rauzy Classes, originally established through translation surface data, by introducing specialized moves that establish sufficiency and necessity.

Contribution

It introduces specialized moves in Rauzy Classes that offer a complete combinatorial proof of the existing classification theorems by Kontsevich, Zorich, and Boissy.

Findings

Complete combinatorial proof of classification theorems

Introduction of specialized moves in Rauzy Classes

Establishment of sufficiency and necessity in classifications

Abstract

Rauzy Classes and Extended Rauzy Classes are equivalence classes of permutations that arise when studying Interval Exchange Transformations. In 2003, Kontsevich and Zorich classified Extended Rauzy Classes by using data from Translation Surfaces, which are associated to IET's thanks to the Zippered Rectangle Construction of Veech from 1982. In 2009, Boissy finalized the classification of Rauzy Classes also using information from Translation Surfaces. We present in this paper specialized moves in (Extended) Rauzy Classes that allows us to prove the sufficiency and necessity in the previous classification theorems. These results provide a complete, and purely combinatorial, proof of these known results. We end with some general statements about our constructed move.