Zapponi-orientable dessins d’enfants

• Ernesto Girondo ^[1] ; Gabino González-Diez ^[1] ; Rubén A. Hidalgo ^[2] ; Gareth A. Jones ^[3]
  1. [1] Universidad Autónoma de Madrid

     Universidad Autónoma de Madrid

     Madrid, España

     
  2. [2] Universidad de La Frontera

     Universidad de La Frontera

     Temuco, Chile

     
  3. [3] University of Southampton

     University of Southampton

     GB.ENG.M4.24UJ, Reino Unido

     

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 2, 2020, págs. 549-570
• Idioma: inglés
• DOI: 10.4171/rmi/1139
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• Resumen

  • 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