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Zbl 1040.05001
Lando, S. K.; Zvonkin, A. K.
(Don B. Zagier)
Graphs on surfaces and their applications. Appendix by Don B. Zagier. (English)
[B] Encyclopaedia of Mathematical Sciences 141(II). Berlin: Springer. xv, 455~p. EUR~101.60 (2004). ISBN 3-540-00203-0/hbk

The goal of this book is to explain the interrelations between three distinct ways to consider an embedded graph: as a topological object, as a sequence of permutations, as a way of representing a ramified covering of the sphere by a compact two-dimensional manifold. \par Chapter 1 introduces the objects of study, namely constellations (finite sequences of permutations), ramified coverings of the sphere, embedded graphs in a variety of forms, and Riemann surfaces. \par Chapter 2 is concerned with dessins d'enfants and the combinatorial and geometric consequences of Belyi's theorem which asserts the faithfulness of a certain group action on maps. \par In Chapter 3 the subject of matrix integrals in map enumeration is introduced. The main emphasis here is the interpretation of the matrix integrals in combinatorial terms as well as the encoding of a combinatorial problem in terms of a matrix integral. \par The method of matrix integrals is a tool to compute the Euler characteristic of moduli spaces of complex algebraic curves, which are the subject of Chapter 4. \par A general study of meromorphic functions is provided in Chapter 5. Enumerative questions are related to algebraic geometry and singularity theory via the Lyashko-Looijenga mapping, while the flexible classification of meromorphic functions is related to braid group action on constellations. \par Chapter 6 explains the structure of the Hopf algebra on chord diagrams interpreted as one vertex maps. \par An appendix by Don Zaiger provides the reader with a short course on representation and character theory of finite groups, with applications to the enumeration of constellations. \par The authors introduce their objects of study, namely graphs and maps, with great care and detail. A wealth of applications is pointed out, but of course these applications are sometimes introduced with minimal or no explanation. The Appendix provides some elegant and concise proofs of results used in the main body of the book and the bibliography contains 313 entries. Well-chosen examples and strategically placed exercises support the reader in gaining an understanding of graphs on surfaces and their fascinating applications.
[Brigitte Servatius (Worcester)]

MSC 2000:: *05-02 Research monographs (combinatorics)
05C10 Topological graph theory
05C30 Enumeration of graphs and maps
14H55 Riemann surfaces
15A52 Random matrices
20Bxx Permutation groups
20F36 Braid groups; Artin groups
30Fxx Riemann surfaces (one complex variable)
32G15 Teichmüller theory
57M15 Relations with graph theory
57M12 Special coverings
57M27 Invariants of knots and 3-manifolds
81T18 Feynman diagrams
81T40 Two-dimensional field theories, etc.

Keywords: constellations; coverings; maps; dessins d'enfants; matrix integrals method; embedded graphs

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