Some remarks on equilateral triangulations of surfaces and Belyi functions

Abstract: In this paper, following Grothendieck {\it Esquisse d'un programme}, which was motivated by Belyi's work, we study some properties of surfaces $X$ which are triangulated by (possibly ideal) isometric equilateral triangles of one of the spherical, euclidean or hyperbolic geometries. These surfaces have a natural Riemannian metric with conic singularities. In the euclidean case we analyze the closed geodesics and their lengths. Such surfaces can be given the structure of a Riemann surface which, considered as algebraic curves, are defined over $\bar{\mathbb{Q}}$ by a theorem of Belyi. They have been studied by many authors of course. Here we define the notion of connected sum of two Belyi functions and give some concrete examples. In the particular case when $X$ is a torus, the triangulation leads to an elliptic curve and we define the notion of a "peel" obtained from the triangulation (which is a metaphor of an orange peel) and relate this peel with the modulus $\tau$ of the elliptic curve. Many fascinating questions arise regarding the modularity of the elliptic curve and the geometric aspects of the Taniyama-Shimura-Weil theory.