Labeled Rauzy classes and framed translation surfaces

TL;DR

This paper compares two definitions of Rauzy classes, relates their differences via coverings, and provides formulas for their degrees and connected components, linking them to moduli spaces of framed translation surfaces.

Contribution

It introduces a formula for the degree of coverings between Rauzy classes and computes the number of connected components of related moduli space coverings.

Findings

Derived a formula for the degree of Rauzy class coverings.

Computed the number of connected components of moduli space coverings.

Connected the labeled Rauzy classes to moduli spaces of framed translation surfaces.

Abstract

In this paper, we compare two definitions of Rauzy classes. The first one was introduced by Rauzy and was in particular used by Veech to prove the ergodicity of the Teichm\"uller flow. The second one is more recent and uses a "labeling" of the underlying intervals, and was used in the proof of some recent major results about the Teichm\"uller flow. The Rauzy diagrams obtained from the second definition are coverings of the initial ones. In this paper, we give a formula that gives the degree of this covering. This formula is related to moduli spaces of "framed" translation surfaces, which corresponds to surfaces where we label horizontal separatrices on the surface. We compute the number of connected component of these natural coverings of the moduli spaces of translation surfaces. Delecroix proved recently a recursive formula for the cardinality of the (reduced) Rauzy classes. Therefore, we also obtain a formula for labeled Rauzy classes.