ABSTRACT
Graphs embedded into surfaces have many important applications, in particular, in combinatorics, geometry, and physics. For example, ribbon graphs and their counting is of great interest in string theory and quantum field theory (QFT). For counting ribbon graphs, we need to know the number of surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group. In Chapter 5, using the restricted congruences studied in Chapter 3, we give an explicit and practical formula for the number of such epimorphisms. As a consequence, we obtain an ‘equivalent‘ form of Harvey's famous theorem on the cyclic groups of automorphisms of compact Riemann surfaces.
