Megamaps: Construction and Examples

Abstract

We consider the usual model of hypermaps or, equivalently, bipartite maps, represented by pairs of permutations that act transitively on a set of edges E. The specific feature of our construction is the fact that the elements of E are themselves (or are labelled by) rather complicated combinatorial objects, namely, the 4-constellations, while the permutations defining the hypermap originate from an action of the Hurwitz braid group on these 4-constellations.The motivation for the whole construction is the combinatorial representation of the parameter space of the ramified coverings of the Riemann sphere having four ramification points.

Keywords

• riemann surface
• ramified covering
• dessins d'enfants
• belyi function
• braid group
• hurwitz scheme
• [info] computer science [cs]
• [math.math-co] mathematics [math]/combinatorics [math.co]
• [info.info-cg] computer science [cs]/computational geometry [cs.cg]
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]