Dessins d'enfants and piecewise-euclidean metrics on Riemann surfaces

There are several ways to introduce such metrics: using Strebel differentials, by metrized ribbon graphs, etc. The talk will contain an overview of these structures and their relation with desssins d'enfants.
According to an old result of the speaker and Voevodsky, a Riemann surface admits a conformal structure, defined by an equilateral triangulation, if and only if the corresponding algebraic curve can be defined over the field of the algebraic numbers. The similar result where the equilateral triangles are replaced by squares, will be presented. As the corresponding dessins d'enfants the square-tiled surfaces (origamis) arise.
Hopefully, soon we'll have a talk by Anton Zorich concerning the statistics of the square-tiled surfaces. It is possible that the relations with the distribution of sizes of their Galois orbits will be found.