Labeled Rauzy classes and 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 Teichmüller 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 Teichmüller 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 [Del10] proved a recursive formula for the cardinality of the (reduced) Rauzy classes. Therefore, we also obtain formula for labeled Rauzy classes.

Cardinality of Rauzy classes

“…Our aim is to count reduced permutations, however in Section 4 we will mainly deal with labeled ones. In [Boi10], C. Boissy analyze the difference between reduced and labeled permutations.…”

“…Their Rauzy diagrams are presented in Figure 3 The labeled rauzy diagrams are coverings of reduced rauzy diagrams (the covering map is the projection (π t , π b ) → π b • π −1 t ). The degree of the covering which gives the multiplicative coefficient between the cardinality of reduced Rauzy classes and labeled Rauzy classes and its computation involves geometric methods which are developed in [Boi10].…”

“…Proof. Following [Vee82] and [Boi10], the permutation σ π (resp. σ π ) can be seen as the crossing of the horizontal direction.…”

“…In this case, the combinatorial data consist in a label for each horizontal outgoing separatrices of the surface. The classification of connected component of this moduli space is done in [Boi10]. In particular, he establishes a formula that relates the cardinality of a labeled Rauzy class of a permutation (π t , π b ) and the cardinality of the reduced Rauzy class of the associated reduced permutation π b • π −1 t .…”

Moduli space of meromorphic differentials with marked horizontal separatrices

“…As in [2], a framed translation surface is a translation surface with a choice, for each singularity of a horizontal separatrix (see Section 3 for a precise definition). When the singularity is a conical singularity (i.e., a zero of the corresponding one-form), it corresponds to a horizontal separatrix.…”