Hypermaps on Surfaces with Boundary

We introduce a theory of hypermaps on surfaces with boundary. A topological hypermap can be associated to an algebraic hypermap which is a quintuple (G, Ω, c1, c2, c3 ,), where G is a group, generated by three involutions c1, c2, and c3 , that acts transitively on the set Ω. Conversely, the topological hypermap can be reconstructed from the algebraic hypermap. This theory is based on the ideas of Cori and Machi, and generalizes the papers of Jones and Singerman and Corn and Singerman.