Ribbon Graphs, Quadratic Differentials on Riemann Surfaces, and Algebraic Curves Defined over $\bar Q$

摘要

It is well known that there is a bijective correspondence between metricribbon graphs and compact Riemann surfaces with meromorphic Strebeldifferentials. In this article, it is proved that Grothendieck's correspondencebetween dessins d'enfants and Belyi morphisms is a special case of thiscorrespondence. For a metric ribbon graph with edge length 1, an algebraiccurve over $\bar Q$ and a Strebel differential on it is constructed. It is alsoshown that the critical trajectories of the measured foliation that isdetermined by the Strebel differential recover the original metric ribbongraph. Conversely, for every Belyi morphism, a unique Strebel differential isconstructed such that the critical leaves of the measured foliation itdetermines form a metric ribbon graph of edge length 1, which coincides withthe corresponding dessin d'enfant.