The Rauzy-Veech induction on the space of interval exchange transformations provides a discretization of the Teichmüller geodesic flow on the moduli space of abelian differentials. Over the last few years, the Rauzy-Veech induction proved to be very powerful in analysing this geodesic flow.
The article under review studies an analogue of the Rauzy-Veech induction for the Teichmüller geodesic flow on the moduli space of quadratic differentials. The role of interval exchange transformations is taken by linear involutions, a notion introduced by C. Danthony and A. Nogueira [C. R. Acad. Sci., Paris, Sér. I 307, No. 8, 409–412 (1988; Zbl 0672.57016)]. Analogously to interval exchange transformations, linear involutions are given by a combinatorial datum (so-called generalized permutations) and continuous data. A generalized permutation is a two-to-one map from a finite set to a set of half cardinality.
The authors develop an algorithm to test whether a given generalized permutation is the combinatorial datum of a linear involution arising from a cross section of the vertical foliation of some quadratic differential. Generalized permutations with this property are called irreducible. They show that the set of irreducible generalized permutations is an attractor for the Rauzy-Veech induction, and moreover, that there is no smaller attractor. More precisely, they prove that the extended Rauzy classes of irreducible generalized permutations are in bijection with the connected component of strata in the moduli spaces of quadratic differentials.