Regular dessins with a given automorphism group

TL;DR

This paper explores the classification and enumeration of regular dessins d'enfants with a specified automorphism group, representing them as quotients of a universal dessin and analyzing their properties and hypermap operations.

Contribution

It provides methods to enumerate all regular dessins with a given automorphism group and represents them as quotients of a universal regular dessin U(G).

Findings

Finite regular dessins with a given automorphism group are finitely many.

All such dessins can be represented as quotients of a single universal dessin U(G).

The genus of U(G) varies significantly depending on the group G.

Abstract

Dessins d'enfants are combinatorial structures on compact Riemann surfaces defined over algebraic number fields, and regular dessins are the most symmetric of them. If G is a finite group, there are only finitely many regular dessins with automorphism group G. It is shown how to enumerate them, how to represent them all as quotients of a single regular dessin U(G), and how certain hypermap operations act on them. For example, if G is a cyclic group of order n then U(G) is a map on the Fermat curve of degree n and genus (n-1)(n-2)/2. On the other hand, if G=A_5 then U(G) has genus 274218830047232000000000000000001. For other non-abelian finite simple groups, the genus is much larger.