Similarity structures on the torus and the Klein bottle via triangulations.

In this paper we study the possibility of defining a similarity structure on the torus and the Klein bottle using the combinatorial data of a triangulation. Given a choice of moduli for the triangles of a triangulation of a surface, the problem is to decide whether such moduli are compatible with a global similarity structure on the surface. We study this problem under two di¤erent viewpoints. From one side we look at the combinatorial data of triangulations, and we develop an algorithmic method, which allows us to reduce the general problem to a simpler one, which is easily solved. From the other side we study the problem more algebraically, looking at the properties of the holonomy, and we give a complete characterization of the choices of moduli defining global similarity structures on the torus (or on the Klein bottle).