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Erschienen in:
2016 | OriginalPaper | Buchkapitel
2. Graph Embeddings
verfasst von : Gareth A. Jones, Jürgen Wolfart
Erschienen in: Dessins d'Enfants on Riemann Surfaces
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Abstract
In this chapter we introduce maps and hypermaps, firstly as topological structures on surfaces, and then as equivalent algebraic objects, described by means of their monodromy groups. We give two further topological and group theoretic definitions of dessins, equivalent to that given in Chap. 1 in terms of Belyĭ functions; these definitions involve bipartite graphs embedded in surfaces, and 2-generator permutation groups. Morphisms, automorphisms and quotients of maps and dessins are defined, together with their regularity properties. Various other possibilities for the graphical representation of dessins are briefly discussed. The chapter includes an instructive example, in which a very simple dessin gives rise to a group of order 95040, the Mathieu group M
12 . The chapter closes with a summary of the finite simple groups, which appear first here and then again in later chapters of this book.