Labeled Rauzy classes and framed translation surfaces

Due to a technical problem several figures were not printed in the 2013 published version. Hence the paper was published a second time in AIF in 2015 (volume 65, no 2)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 proved recently a recursive formula for the cardinality of the (reduced) Rauzy classes. Therefore, we also obtain a formula for labeled Rauzy classes