Revista Matemática Iberoamericana


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Published online first: 2019-12-17
DOI: 10.4171/rmi/1139

Zapponi-orientable dessins d’enfants

Ernesto Girondo[1], Gabino González-Diez[2], Rubén A. Hidalgo[3] and Gareth A. Jones[4]

(1) Universidad Autónoma de Madrid, Spain
(2) Universidad Autónoma de Madrid, Spain
(3) Universidad de La Frontera, Temuco, Chile
(4) University of Southampton, UK

Almost two decades ago, Zapponi introduced a notion of orientability of a clean dessin d’enfant, based on an orientation of the embedded bipartite graph. We extend this concept, which we call Z-orientability to distinguish it from the traditional topological definition, to the wider context of all dessins, and we use it to define a concept of twist orientability, which also takes account of the Z-orientability properties of those dessins obtained by permuting the roles of white and black vertices and face-centres. We observe that these properties are Galois-invariant, and we study the extent to which they are determined by the standard invariants such as the passport and the monodromy and automorphism groups. We find that in general they are independent of these invariants, but in the case of regular dessins they are determined by the monodromy group.

Keywords: Dessins d’enfants, bipartite graphs, Belyi functions, Galois invariants

Girondo Ernesto, González-Diez Gabino, Hidalgo Rubén, Jones Gareth: Zapponi-orientable dessins d’enfants. Rev. Mat. Iberoam. Electronically published on December 17, 2019. doi: 10.4171/rmi/1139 (to appear in print)