Dihedral and cyclic symmetric maps on surfaces

Abstract

If the face\mbox{-}cycles at all the vertices in a map are of the same type, then the map is said to be a semi-equivelar map. Automorphism (symmetry) of a map can be thought of as a permutation of the vertices which preserves the vertex\mbox{-}edge\mbox{-}face incidences in the embedding. The set of all symmetries forms the symmetry group. In this article, we discuss the maps' symmetric groups on higher genus surfaces. In particular, we show that there are at least $39$ types of the semi-equivelar maps on the surface with Euler char. $-2m, m \ge 2$ and the symmetry groups of the maps are isomorphic to the dihedral group or cyclic group. Further, we prove that these $39$ types of semi-equivelar maps are the only types on the surface with Euler char. $-2$. Moreover, we know the complete list of semi-equivelar maps (up to isomorphism) for a few types. We extend this list to one more type and can classify others similarly. We skip this part in this article.