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I was recommended this forum to be the leading site for algebraic geometry, so I would like to ask you a question about Grothendieck dessins d´enfants. My background is in maps on surfaces (graph embeddings), and I got fascinated about their application to dessins d´enfants.

As an example, I found it completely stunning when I saw the first time that the least genus embedding of $K_{n,n,n}$ (the complete tripartite graph with $3n$ vertices), when viewed as a dessin d'enfants, specifies the Fermat curve. (cf. Jones & Singermann "Maps, Hypermaps, and Triangle Groups" in: Schneps (ed.): The Grothendieck Theory of Dessins d’Enfants. London Mathematical Society Lecture Notes Series 200, Cambridge University Press, Cambridge 1994)

So I would like to look for more areas of potential application of graph embeddings to dessins d'enfants, but don't have a good overview yet over recent developments in this field.

Would you be able to recommend a recent survey of dessins d'enfants, or a good recent text to me?

I am not sure whether I am using the right tags for this, please feel free to adjust!

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    $\begingroup$ I don't know to what extent it fits your criteria, but it's probably worth having a look at the book by Ernesto Girondo and Gabino González-Diez, ‘Introduction to Compact Riemann Surfaces and Dessins d'Enfants’ (LMS Student Texts 79 (2012)). $\endgroup$
    – Gro-Tsen
    Commented Jul 30, 2020 at 10:29
  • $\begingroup$ @Gro-Tsen Merci beaucoup! Very grateful for your suggestion! As I am new to this forum, I am not sure how to give you upvote credit for your help? Would be happy to express my thanks $\endgroup$
    – gos Hélène
    Commented Jul 30, 2020 at 13:52
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    $\begingroup$ Szamuely's "Galois Groups and Fundamental Groups" sections 4.6-4.9 are a great background to the subject matter, but they don't go into much detail about the dessins themselves. $\endgroup$
    – PrimeRibeyeDeal
    Commented Jul 30, 2020 at 17:51
  • $\begingroup$ @PrimeRibeyeDeal Thank you very much for your recommendation! Very interested in that $\endgroup$
    – gos Hélène
    Commented Jul 31, 2020 at 6:11
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    $\begingroup$ See Jones & Wolfart, "Dessins d'Enfants on Riemann Surfaces" (Springer, 2016). $\endgroup$
    – Tom Harris
    Commented Aug 5, 2020 at 9:03

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