“Regular Dessins with a Given Automorphism Group” by Gareth A. Jones

Gareth A. Jones has a new paper on the ArXiv entitled “Regular Dessins with a Given Automorphism Group”.

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.