A regular dessin d'enfant, in this paper, will be a pair (S,β), where S is a closed Riemann surface and β: S → ℂ̂ is a regular branched cover whose branch values are contained in the set {∞,0,1}. 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 ⋊ ℤ₂, then (S,β) is also definable over ℚ. Moreover, if A ≅ ℤₙ, then we provide explicitly these dessins over ℚ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-3-3,
author = {Ruben A. Hidalgo},
title = {Almost Abelian regular dessins d'enfants},
journal = {Fundamenta Mathematicae},
volume = {220},
year = {2013},
pages = {269-278},
zbl = {1325.14051},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-3-3}
}
Ruben A. Hidalgo. Almost Abelian regular dessins d'enfants. Fundamenta Mathematicae, Tome 220 (2013) pp. 269-278. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-3-3/