Classification of thin regular map representations of hypermaps

Abstract

There are two well known maps representations of hypermaps, namely the Walsh and the Vince map representations, being dual of each other. They correspond to normal subgroups of index two of a free product Γ = (C2×C2) * C2 which decompose as “elementary” free product C2 * C2 * C2. However Γ has three normal subgroups that decompose as “elementary" free product C2 * C2 * C2, the third of these sbgroups giving the less known petrie-path map representation. By relaxing the “elementary" free product condition to free product of rank 3, and under the extra condition “words of smaller length" on the generators, we prove that the number of map representations of hypermaps increases to 15 (up to a restrictedly dual), all of which described in this paper.