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Archiv der Mathematik

, Volume 112, Issue 1, pp 13–18 | Cite as

Automorphism groups of dessins d’enfants

  • Rubén A. HidalgoEmail author
Article

Abstract

Recently, Gareth Jones observed that every finite group G can be realized as the group of automorphisms of some dessin d’enfant \({{\mathcal {D}}}\). In this paper, complementing Gareth’s result, we prove that for every possible action of G as a group of orientation-preserving homeomorphisms on a closed orientable surface of genus \(g \ge 2\), there is a dessin d’enfant \({{\mathcal {D}}}\) admitting G as its group of automorphisms and realizing the given topological action. In particular, this asserts that the strong symmetric genus of G is also the minimum genus action for it to act as the group of automorphisms of a dessin d’enfant of genus at least two.

Keywords

Riemann surfaces Dessins d’enfants Automorphisms 

Mathematics Subject Classification

Primary 30F40 11G32 14H57 

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Notes

Acknowledgements

The author is very grateful to Gareth Jones for discussions about his preprint [6], which motivated this work, and to Ernesto Girondo and Gabino González-Diez for the many discussions about dessins d’enfants. Also, the author would like to thank the referee for the valuable corrections/suggestions and by pointing out Remark 1.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Matemática y EstadísticaUniversidad de La FronteraTemucoChile

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