Encyclopaedia of Mathematical Sciences 141(II). Berlin: Springer (ISBN 3-540-00203-0/hbk). xv, 455 p. EUR 101.60 (2004).
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.
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.
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.
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.
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.
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.
Chapter 6 explains the structure of the Hopf algebra on chord diagrams interpreted as one vertex maps.
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.
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.