Regular hypermaps

In this thesis we use triangle groups and their subgroups to investigate properties of hypermaps. The first two chapters review the more well-known theory of maps on compact orientable surfaces without boundary. After outlining the basic properties of algebraic and topological hypermaps we develop a method, in chapter four, for the construction of maps and hypermaps both regular and irregular, based on properties of Schreier generators. These enable us to pair sides in fundamental regions obtaining the map or hypermap sought. In chapter five we describe all Universal hypermaps that lie on simply-connected surfaces before undertaking an outline of the connection between maps and hypermaps in chapter six. We demonstrate that, in a loose sense, maps are special cases of hypermaps whereas hypermaps can be regarded as particular types of maps. In this regard we show that regular hypermaps on the torus correspond exactly to regular maps. In chapter seven we construct many examples of such hypermaps including a new description of the genus 1 imbedding of the projective plane of order 2. Finally we classify and illustrate all regular hypermaps of genus 2 utilising the methods outlined earlier.