Embedding of the Suborbital Graphs corresponding to the Normalizer

Abstract

It is known that an embedding of a graph ℱ on a surface corresponds to a representation of ℱ on , where the edges of ℱ splits into polygonal cells while the vertices of ℱ exactly correspond to the punctures of . Let ℱ be embedded on a surface . The complement of vertices and edges on forms a family of regions called faces. If each of these faces is homeomorphic to an open disk then the embedding is called a map, and if the map has identical regular polygonal faces, then it is called regular. Indeed the regular maps are the natural generalizations of the well-known platonic solids. This study is devoted to investigating the embedding of the graphs arising from the action of the normalizer of Γ in 2, , which acquire significance since the investigation of the monstrous moonshine. We define a suitable chain of map subgroups of the normalizer and study the regular maps corresponding to these subgroups.