Regular tessellations of surfaces and (p, q, 2)-triangle groups

The purpose of this paper is a study of the existence and classification of certain regular tessellations of surfaces and their relationship with the Schwarz triangle groups of type (p, q, 2). These tessellations are the natural generalizations to arbitrary surfaces of the spherical tessellations corresponding to the five Platonic Solids. A regular tessellation of type {p, q}, a {p, q}-pattern, for short, on a surface M is a tessellation of M by p-sided faces such that the valence of each vertex is q. (A more precise definition appears in ? 1.) No global symmetry is assumed. A { p, q}-pattern on M is said to be geometric if M admits a Riemannian metric of constant curvature with respect to which the edges of the pattern are geodesic arcs of equal length and the interior angles at all vertices in all faces are (2 7/q). If the curvature happens to he zero then we further insist that the area of each face be 1. It is natural to call two {p, q}-patterns on M