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Introduction to Compact Riemann Surfaces and Dessins d'Enfants Paperback – Feb 20 2012

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"Overall the text is very well written and easy to follow, partly due to the abundance of good concrete examples in every single section illustrating concepts from the very basic to the very technical."
Aaron D. Wootton, Mathematical Reviews

Book Description

Starting with a friendly account of the theory of compact Riemann surfaces, this 2011 book then introduces the Belyi-Grothendieck theory of dessins d'enfants, taking the reader with no previous knowledge of the subject to the forefront of current research. Many worked examples and illustrations are provided.

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extremely complete, intuitive, and readable April 26 2015
By Sean Rostami - Published on
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This is quite an excellent book, and very much worth purchasing (if you are interested at all in this subject). Really, it is the only book of its kind, as all other books/papers that I know of, which are also valuable for different reasons, are either collections of more specialized articles (e.g. Schneps) or, at the other extreme, have very little technical depth. But even if there were other competing books, this would likely still be the best, since it is unbelievably engaging and easy-to-read. You really can start at page 1 and effortlessly read the whole book, acquiring a very firm understanding of the subject by the end. The subject is, by nature, very visual and the authors include a large number of clear pictures and diagrams to accompany the exposition.

In case the authors are reading, here are a few things that I suggest for future printings/editions:
(1) Include some basic treatment of the Grothendieck-Teichmuller group, which could be done in an elementary way as in Pierre Guillot's preprint arXiv:1309.1968. This could maybe be merged with section 4.3.1, which comments on the monodromy action coming from a "de-ramified" Belyi map.
(2) Before Proposition 3.3, it might be nice to mention that Belyi's special polynomials are themselves Belyi maps whose dessins are "double stars" (read near Figure 4.21).
(3) In section 3.1, in the middle of the proof: Is that equality involving B_1 really an equality?
(4) In the proof of Proposition 4.13, it might be nice to make additional comments on what to do while "us[ing] \sigma_0 to label the remaining edges" when the edge in question is fixed by \sigma_0.
(5) Before Example 4.14, there is a typo: the 2nd \sigma_0 on the right-hand-side should be replaced with a \sigma_1.
(6) In Proposition 4.20, I think it is somewhat misleading to say that there is a function {Dessins} to {Belyi Pairs}. It seems to me that there is only a function at the level of equivalence classes. In the opposite direction, {Belyi Pairs} to {Dessins}, there really is a function: color the inverse image of [0,1].
(7) Remark 4.26: It might be nice to expand this topic (weakening the definition of equivalence to allow transformation of the codomain) into a full subsection.
(8) Example 4.58: For the convenience of the reader, it might be nice to include the field splitting the cubic and its galois group (which are, respectively, the field over Q generated by cubic root of 2 and the symmetric group S3).
(9) Example 4.24: Maybe I misunderstand the meaning of the labels of Figure 4.7, but I think some of the numbers in the "sheets" are missing signs (e.g. the point marked "2" should be marked "-2"). This may happen in other example also.