Maps of Archimedean class and operations on dessins

Abstract

In the present paper we introduce a family of functors (called operations) of the category of hypermaps (dessins) preserving the underlying Riemann surface. The considered family of functors include as particular instances the operations considered by Magot and Zvonkin (2000), Singerman and Syddall (2003), and Girondo (2003). We identify a set of 10 operations in the above infinite family which produce vertex-transitive dessins out of regular ones. This set is complete in the following sense: if a vertex-transitive map arises from a regular dessin H applying an operation, then it can be obtained from a regular dessin on the same surface (possibly different from H) applying one of the 10 operations. The statement includes the classical case when the underlying surface is the sphere.