Abstract
An algebraic map is a pair ( G , Ω), where G is a group generated by x , y , with x 2 = 1, acting transitively on a set Ω. It is regular if its automorphism group is transitive on Ω. Out of an algebraic map we can construct a topological map which is a two-cell decomposition of an orientable surface. There has been a lot of work on regular maps; on the sphere, for example, they are the Platonic solids. In this paper we show how we can construct a topological hypermap out of an algebraic hypermap. We put particular emphasis on the regular ones. On the sphere they are Archimedean solids and we also describe all examples on a torus and on a surface of genus 2.
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