Almost Abelian regular dessins d'enfants

Abstract

A regular dessin d'enfant, in this paper, will be a pair (S; β), where S is a closed Riemann surface and β: S → bC is a regular branched cover whose branch values are contained in the set f1; 0; 1g. Let Aut(S; β) be the group of automorphisms of (S; β), that is, the deck group of β. If Aut(S; β) is Abelian, then it is known that (S; β) can be defined over Q. We prove that, if A is an Abelian group and Aut(S; β) ≅= A o Z2, then (S; β) is also definable over Q. Moreover, if A ≅= Zn, then we provide explicitly these dessins over Q. © 2013 Instytut Matematyczny PAN.