A THEORY OF RESTRICTED REGULARITY OF HYPERMAPS

A THEORY OF RESTRICTED REGULARITY OF HYPERMAPS

Hypermaps are cellular embeddings of hypergraphs in compact and connected surfaces, and are a generalisation of maps, that is, 2-cellular decompositions of closed surfaces. There is a well known correspondence between hypermaps and co-compact subgroups of the free product $\Delta=C_2*C_2*C_2$. In this correspondence, hypermaps correspond to conjugacy classes of subgroups of $\Delta$, and hypermap coverings to subgroup inclusions. Towards the end of [9] the authors studied regular hypermaps with extra symmetries, namely, G-symmetric regular hypermaps for any subgroup G of the outer automorphism Out$(\Delta)$ of the triangle group $\Delta$. This can be viewed as an extension of the theory of regularity. In this paper we move in the opposite direction and restrict regularity to normal subgroups $\Theta$ of $\Delta$ of finite index. This generalises the notion of regularity to some non-regular objects.