Graph Embeddings

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

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.  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.

Keywords

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

References

1. 1.
Aschbacher, M.: Sporadic Groups. Cambridge University Press, Cambridge (1994)
2. 2.
Bryant, R.P., Singerman, D.: Foundations of the theory of maps on surfaces with boundary. Q. J. Math. (2) 36, 17–41 (1985)Google Scholar
3. 3.
Cameron, P.J., van Lint, J.H.: Designs, Graphs, Codes and Their Links. London Mathematical Society Student Texts, vol. 22. Cambridge University Press, Cambridge (1991)Google Scholar
4. 4.
Carter, R.W.: Simple Groups of Lie Type. Wiley, London/New York/Sydney (1972)
5. 5.
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: ATLAS of Finite Groups. Clarendon Press, Oxford (1985)
6. 6.
Cori, R.: Un code pour les graphes planaires et ses applications. In: Astérisque, vol. 27. Société Mathématique de France, Paris (1975)Google Scholar
7. 7.
Cori, R., Machì, A.: Maps, hypermaps and their automorphisms: a survey, I, II, III. Expo. Math. 10, 403–427, 429–447, 449–467 (1992)Google Scholar
8. 8.
Dixon, J.D., Mortimer, B.: Finite Permutation Groups. Graduate Texts in Mathematics, vol. 163. Springer, Berlin/Heidelberg/New York (1996)Google Scholar
9. 9.
Fulton, W., Harris, J.: Representation Theory. Springer, Berlin/Heidelberg/New York (1991)
10. 10.
Hamilton, W.R.: Account of the icosian calculus. Proc. R. Ir. Acad. 6, 415–416 (1858). In: Halberstam, H., Ingram, R.E. (eds.) The Mathematical Papers of Sir William Rowan Hamilton, Vol. III Algebra, p. 609. Cambridge University Press, Cambridge (1967)Google Scholar
11. 11.
Hamilton, W.R.: Letter to John T. Graves on the icosian. In: Halberstam, H., Ingram, R.E. (eds.) The Mathematical Papers of Sir William Rowan Hamilton, Vol. III Algebra, pp. 612–625. Cambridge University Press, Cambridge (1967)Google Scholar
12. 12.
James, L.D.: Operations on hypermaps and outer automorphisms. Eur. J. Comb. 9, 551–560 (1988)
13. 13.
Jones, G.A.: Graph embeddings, groups and Riemann surfaces. In: Algebraic Methods in Graph Theory I, II, Szeged 1978. Colloqium of Mathematical Society János Bolyai, vol. 25. North-Holland, Amsterdam (1981)Google Scholar
14. 14.
Jones, G.A., Jones, J.M., Wolfart, J.: On the regularity of maps. J. Comb. Theory Ser. B 98, 631–636 (2008)
15. 15.
Liebeck, M., Praeger, C., Saxl, J.: A classification of the maximal subgroups of the finite alternating and symmetric groups. J. Algebra 111, 365–383 (1987)
16. 16.
Malle, G.: Fields of definition of some three point ramified field extensions. In: Schneps, L. (ed.) The Grothendieck Theory of Dessins d’Enfants. London Mathematical Society Lecture Note Series, vol. 200, pp. 147–168. Cambridge University Press, Cambridge (1994)
17. 17.
McMullen, P., Schulte, E.: Abstract Regular Polytopes. Cambridge University Press, Cambridge (2002)
18. 18.
Serre, J.-P.: Complex Semisimple Lie Algebras. Springer, Berlin/Heidelberg/New York (1987)
19. 19.
The GAP Group: GAP – Groups, Algorithms, and Programming. Version 4.7.6 (2014). http://www.gap-system.org. Accessed 20 January 2015
20. 20.
Tutte, W.T.: What is a map? In: New Directions in the Theory of Graphs, Ann Arbor, 1971, pp. 309–325. Academic, New York (1973)Google Scholar
21. 21.
Voisin, C., Malgoire, J.: Cartes cellulaires. In: Cahiers Mathématiques, vol. 12. Université de Montpellier, Montpellier (1977)Google Scholar
22. 22.
Walsh, T.R.S.: Hypermaps versus bipartite maps. J. Comb. Theory Ser. B 18, 155–163 (1975)
23. 23.
Wilson, R.A.: The Finite Simple Groups. Springer, London (2009)

© 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