Abstract
We develop a Belyi type theory that applies to Klein surfaces, i.e. (possibly non-orientable) surfaces with boundary which carry a dianalytic structure. In particular we extend Belyi’s famous theorem from Riemann surfaces to Klein surfaces.
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References
Alling N.L., Greenleaf N.: Foundations of the Theory of Klein Surfaces. Springer Verlag, Berlin (1971).
Belyĭ G.V.: On Galois extensions of a maximal cyclotomic field. Izv. Akad. Nauk SSSR Ser. Mat. 43, 267–276, 479 (1979).
Biggs N.: The symplectic representation of map automorphisms. Bull. London Math. Soc. 4, 303–306 (1972).
Bryant R.P., Singerman D.: Foundations of the theory of maps on surfaces with boundary. Q. J. Math. Oxford II. 36, 17–41 (1985).
Bryant R.P.: Maps on surfaces with boundary. Ph.D. thesis, University of Southampton (1984).
Cohen P.B., Itzykson C., Wolfart J.: Fuchsian triangle groups and Grothendieck dessins. Variations on a theme of Belyi. Commun. Math. Phys. 163, No. 3, 605–627 (1994).
Cori R.: Un Code pour les graphes planaires et ses applications. Astérisque 27, 1–169, Soc. Math. France, Paris (1975).
Corn D., Singerman D.: Regular hypermaps. Eur. J. Comb. 9, No.4, 337–351 (1988).
Corn D.: Regular hypermaps. Ph.D.thesis, University of Southampton (1989).
Coxeter H.S.M., Moser W.O.J.: Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, Berlin / Heidelberg / New York (1980).
Grothendieck A.: Esquisse d’un Programme. Geometric Galois Actions I, Around Grothendieck’s Esquisse d’un Programme, Lochak P., Schneps L. (eds.) London Math. Soc. Lecture Note Ser. 242, 5–48, Cambridge University Press, Cambridge (1997).
Izquierdo M., Singerman D.: Hypermaps on surfaces with boundary. Eur. J. Comb. 15, No.2, 159–172 (1994).
Jones G.A., Singerman D.: Belyĭ functions, hypermaps and Galois groups. Bull. London Math. Soc. 28, 561–590 (1996).
Jones G. A., Singerman D.: Maps, hypermaps and triangle groups. The Grothendieck Theory of Dessins d’Enfants, Schneps L. (ed.), London Math. Soc. Lecture Note Ser. 200, 115–145, Cambridge University Press, Cambridge (1994).
Jones G.A., Singerman D.: Theory of maps on orientable surfaces. Proc. London Math. Soc. (3) 37, 273–307 (1978).
Jones G.A.: Maps on surfaces and Galois groups. Math.Slovaca 47,1–33 (1997).
Klein F.: Über die Transformationen siebenter Ordnung der elliptischen Funktionen. Math. Ann. 14, 428–471 (1879).
Köck B., Singerman D.: Real Belyi theory. Q. J. Math. 58, 463–478 (2007).
Köck B.: Belyi’s theorem revisited. Beitr. Algebra Geom. 45, No. 1, 253–265 (2004).
Levy S.: The eightfold way. The beauty of Kleins quartic curve. Mathematical Sciences Research Institute Publications 35, Cambridge University Press, Cambridge (2001).
Singerman, D.: Finitely maximal Fuchsian groups. J. London Math. Soc. (2) 6, 29–38 (1972).
Singerman D.: Automorphisms of maps, permutation groups and Riemann surfaces. Bull. Lond. Math. Soc. 8, 65–68 (1976).
Tutte W.T.: What is a map? New directions in graph theory, Harary F. (ed.), 309–325, Academic Press (1973).
Walsh T.R.S.: Hypermaps versus bipartite maps. J. Combin. Theory Ser. B 18, 155–163 (1975).
Wolfart J.: The ‘obvious’ part of Belyis theorem and Riemann surfaces with many automorphisms. Schneps L. (ed.), Geometric Galois actions. 1. Around Grothendiecks “Esquisse dun programme”. Proceedings of the conference on geometry and arithmetic of moduli spaces, Luminy, France, August 1995. Lond. Math. Soc. Lect. Note Ser. 242, 97–112, Cambridge University Press, Cambridge (1997).
Zvonkin, A.: Megamaps: Construction and examples. Discrete models: combinatorics, computation, and geometry. Proceedings of the 1st international conference (DM-CCG), Paris, France, July 2-5, 2001. Discrete Math. Theor. Comput. Sci., Proc. AA, 329–340, Maison de l’Informatique et des Mathématiques Discrètes, Paris (2001).
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Singerman, D. (2016). Triangle Groups and Maps. In: Širáň, J., Jajcay, R. (eds) Symmetries in Graphs, Maps, and Polytopes. SIGMAP 2014. Springer Proceedings in Mathematics & Statistics, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-30451-9_16
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DOI: https://doi.org/10.1007/978-3-319-30451-9_16
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