Graph Embeddings

  • Gareth A. Jones
  • Jürgen Wolfart
Part of the Springer Monographs in Mathematics book series (SMM)


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 M12 . The chapter closes with a summary of the finite simple groups, which appear first here and then again in later chapters of this book.


Automorphism group Bipartite map Dessin d’enfant Hypermap Map Mathieu group Monodromy group Regular dessin Simple group 


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Gareth A. Jones
    • 1
  • Jürgen Wolfart
    • 2
  1. 1.School of MathematicsUniversity of SouthamptonSouthamptonUK
  2. 2.Johann Wolfgang Goethe-UniversitätFrankfurt am MainGermany

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