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Beauville Surfaces and Groups: A Survey

  • Gareth A. JonesEmail author
Chapter
Part of the Fields Institute Communications book series (FIC, volume 70)

Abstract

This is a survey of recent progress on Beauville surfaces, concentrating almost entirely on the group-theoretic and combinatorial problems associated with them. A Beauville surface \(\mathcal{S}\) is a complex surface formed from two orientably regular hypermaps of genus at least 2 (viewed as compact Riemann surfaces and hence as algebraic curves), with the same automorphism group G acting freely on their product. The following questions are discussed: Which groups G (called Beauville groups) have this property? What can be said about the automorphism group and the fundamental group of \(\mathcal{S}\)? Beauville surfaces are defined (as algebraic varieties) over the field \(\overline{\mathbb{Q}}\) of algebraic numbers, so how does the absolute Galois group \(\mathrm{Gal}\,\overline{\mathbb{Q}}/\mathbb{Q}\) act on them?

Keywords

Beauville surface Beauville group Hypermap Compact Riemann surface Absolute Galois group 

Subject Classifications

Primary 20B25 Secondary 05C10 11G32 14J25 14J50 51M20 

Notes

Acknowledgements

The author is grateful to Gabino González-Diez and Bernhard Köck for some very helpful comments on an earlier draft of this paper, and to the organisers of the Workshop on Symmetry in Graphs, Maps and Polytopes at the Fields Institute, Toronto, 24–27 October 2011, for the opportunity to give a talk on which it is based.

References

  1. 1.
    Barker, N., Boston, N., Fairbairn, B.: A note on Beauville p-groups. Exp. Math. 21, 298–306 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Barker, N., Boston, N., Peyerimhoff, N., Vdovina, A.: New examples of Beauville surfaces. Monatsh. Math. 166, 319–327 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bauer, I., Catanese, F., Grunewald, F.: Beauville surfaces without real structures I. In: Bogomolov, F., Tschinkel, Y. (eds.) Geometric Methods in Algebra and Number Theory. Progress in Mathematics, vol. 235, pp. 1–42. Birkhäuser, Boston (2005)CrossRefGoogle Scholar
  4. 4.
    Bauer, I., Catanese, F., Grunewald, F.: Chebycheff and Belyi polynomials, dessins d’enfants, Beauville surfaces and group theory. Mediterr. J. Math. 3, 121–146 (2006)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Beauville, A.: Surfaces Algébriques Complexes. Astérisque, vol. 54. Société Mathématique de France, Paris (1978)Google Scholar
  6. 6.
    Belyĭ, G.V.: On Galois extensions of a maximal cyclotomic field. Math. USSR Izv. 14, 247–256 (1980)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bridson, M.R., Conder, M.D.E., Reid, A.W.: Determining Fuchsian groups by their finite quotients (submitted)Google Scholar
  8. 8.
    Carter, R.W.: Simple Groups of Lie Type. Wiley, London/New York/Sydney (1972)zbMATHGoogle Scholar
  9. 9.
    Catanese, F.: Fibred surfaces, varieties isogenous to a product and related moduli spaces. Am. J. Math. 122, 1–44 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Conder, M.D.E.: Hurwitz groups with arbitrarily large centres. Bull. Lond. Math. Soc. 18, 269–271 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: ATLAS of Finite Groups. Clarendon, Oxford (1985)zbMATHGoogle Scholar
  12. 12.
    Coste, A.D., Jones, G.A., Streit, M., Wolfart, J.: Generalised Fermat hypermaps and Galois orbits. Glasg. Math. J. 51, 289–299 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Fairbairn, B.T., Magaard, K., Parker, C.W.: Generation of finite simple groups with an application to groups acting on Beauville surfaces. arXiv:math.GR/1010.3500Google Scholar
  14. 14.
    Fuertes, Y., González-Diez, G.: On Beauville structures on the groups S n and A n. Math. Z. 264, 959–968 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Fuertes, Y., González-Diez, G.: On the number of automorphisms of unmixed Beauville surfaces (preprint)Google Scholar
  16. 16.
    Fuertes, Y., González-Diez, G., Jaikin-Zapirain, A.: On Beauville surfaces. Groups Geom. Dyn. 5, 107–119 (2011)CrossRefzbMATHGoogle Scholar
  17. 17.
    Fuertes, Y., Jones, G.A.: Beauville surfaces and finite groups. J. Algebra 340, 13–27 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Garion, S., Larsen, M., Lubotzky, A.: Beauville surfaces and finite simple groups. J. Reine Angew. Math. 666, 225–243 (2012)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Garion, S., Penegini, M.: Beauville surfaces, moduli spaces and finite groups (2011). arXiv:math.AG/1107.5534v1Google Scholar
  20. 20.
    Girondo, E., González-Diez, G.: A note on the action of the absolute Galois group on dessins. Bull. Lond. Math. Soc. 39, 721–723 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Girondo, E., González-Diez, G.: Introduction to Compact Riemann Surfaces and Dessins d’Enfants. London Mathematical Society Student Texts, vol. 79. Cambridge University Press, Cambridge (2012)Google Scholar
  22. 22.
    González-Diez, G.: Belyi’s theorem for complex surfaces. Am. J. Math. 130, 59–74 (2008)CrossRefzbMATHGoogle Scholar
  23. 23.
    González-Diez, G., Jaikin-Zapirain, A.: The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces (submitted)Google Scholar
  24. 24.
    González-Diez, G., Jones, G.A., Torres-Teigell, D.: Beauville surfaces with abelian Beauville group. Math. Scand. (in press)Google Scholar
  25. 25.
    González-Diez, G., Jones, G.A., Torres-Teigell, D.: Arbitrarily large Galois orbits of non-homeomorphic surfaces. arXiv:math.AG/1110.4930Google Scholar
  26. 26.
    González-Diez, G., Torres-Teigell, D.: Non-homeomorphic Galois conjugate Beauville surfaces of minimum genera. Adv. Math. 229, 3096–3122 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    González-Diez, G., Torres-Teigell, D.: An introduction to Beauville surfaces via uniformization. In: Jiang, Y., Mitra, S. (eds.) Quasiconformal Mappings, Riemann Surfaces, and Teichmüler Spaces. Contemporary Mathematics, vol. 575, pp. 123–151. American Mathematical Society, Providence (2012)CrossRefGoogle Scholar
  28. 28.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1978)zbMATHGoogle Scholar
  29. 29.
    Grothendieck, A.: Esquisse d’un programme. In: Lochak, P., Schneps, L. (eds.) Geometric Galois Actions 1. Around Grothendieck’s Esquisse d’un Programme. London Mathematical Society Lecture Note Series, vol. 242, pp. 5–84. Cambridge University Press, Cambridge (1997)Google Scholar
  30. 30.
    Guralnick, R., Malle, G.: Simple groups admit Beauville structures. J. Lond. Math. Soc. 85(2), 694–721 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Havas, G., Wall, G.E., Wamsley, J.W.: The two generator restricted Burnside group of exponent five. Bull. Aust. Math. Soc. 10, 459–470 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Hurwitz, A.: Über algebraische Gebilde mit eindeutigen Transformationen in sich. Math. Ann. 41, 403–442 (1893)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Jones, G.A.: Regular embeddings of complete bipartite graphs: classification and enumeration. Proc. Lond. Math. Soc. 101(3), 427–453 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Jones, G.A.: Automorphism groups of Beauville surfaces. J. Group Theory 16, 353–381 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Jones, G.A., Jones, J.M.: Elementary Number Theory. Springer Undergraduate Mathematics Series. Springer, London (1998)CrossRefzbMATHGoogle Scholar
  36. 36.
    Jones, G.A., Nedela, R., Škoviera, M.: Regular embeddings of K n, n where n is an odd prime power. Eur. J. Comb. 28, 1863–1875 (2007)CrossRefzbMATHGoogle Scholar
  37. 37.
    Jones, G.A., Singerman, D.: Belyi functions, hypermaps and Galois groups. Bull. Lond. Math. Soc. 28, 561–590 (1996)CrossRefMathSciNetGoogle Scholar
  38. 38.
    Jones, G.A., Singerman, D.: Maps, hypermaps and triangle groups. In: Schneps, L. (ed.) The Grothendieck Theory of Dessins d’enfants (Luminy, 1993). London Mathematical Society Lecture Note Series, vol. 200, pp. 115–145. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
  39. 39.
    Jones, G.A., Streit, M.: Galois groups, monodromy groups and cartographic groups. In: Lochak, P., Schneps, L. (eds.) Geometric Galois Actions 2. The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups. London Mathematical Society Lecture Note Series, vol. 243, pp. 25–65. Cambridge University Press, Cambridge (1997)Google Scholar
  40. 40.
    Jones, G.A., Streit, M., Wolfart, J.: Galois action on families of generalised Fermat curves. J. Algebra 307, 829–840 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Jones, G.A., Streit, M., Wolfart, J.: Wilson’s map operations on regular dessins and cyclotomic fields of definition. Proc. Lond. Math. Soc. 100, 510–532 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Köck, B.: Belyi’s theorem revisited. Beiträge zur Algebra und Geometrie 45, 253–275 (2004)zbMATHGoogle Scholar
  43. 43.
    Kostrikin, A.I.: The Burnside problem (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 23, 3–34 (1959)zbMATHMathSciNetGoogle Scholar
  44. 44.
    Lucchini, A.: (2, 3, k)-generated groups of large rank. Arch. Math. 73, 241–248 (1999)Google Scholar
  45. 45.
    Macbeath, A.M.: Generators of the linear fractional groups. In: Leveque, W.J., Straus, E.G. (eds.) Number Theory. Proceedings of Symposia in Pure Mathematics, Houston 1967, vol. 12, pp. 14–32. American Mathematical Society, Providence (1969)Google Scholar
  46. 46.
    Malle, G., Matzat, B.H.: Inverse Galois Theory. Springer Monographs in Mathematics. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  47. 47.
    O’Brien, E.A., Vaughan-Lee, M.: The 2-generator restricted Burnside group of exponent 7. Int. J. Algebra Comput. 12, 575–592 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Reyssat, E.: Quelques Aspects des Surfaces de Riemann. Birkhäuser, Boston/Basel/Berlin (1989)zbMATHGoogle Scholar
  49. 49.
    Schneps, L.: Dessins d’enfants on the Riemann sphere. In: Schneps, L. (ed.) The Grothendieck Theory of Dessins d’enfants (Luminy, 1993). London Mathematical Society Lecture Note Series, vol. 200, pp. 47–77. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
  50. 50.
    Serre, J.-P.: Variétées projectives conjuguées non homéomorphes. Comptes Rendues Acad. Sci. Paris 258, 4194–4196 (1964)zbMATHGoogle Scholar
  51. 51.
    Serre, J.-P.: Topics in Galois Theory. Jones and Bartlett, Boston (1992)zbMATHGoogle Scholar
  52. 52.
    Singerman, D.: Finitely maximal Fuchsian groups. J. Lond. Math. Soc. 6(2), 29–38 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  53. 53.
    Streit, M.: Field of definition and Galois orbits for the Macbeath-Hurwitz curves. Arch. Math. 74, 342–349 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    Streit, M., Wolfart, J.: Characters and Galois invariants of regular dessins. Rev. Mat. Complut. 13, 49–81 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  55. 55.
    Streit, M., Wolfart, J.: Cyclic projective planes and Wada dessins. Documenta Mathematica 6, 39–68 (2001)MathSciNetGoogle Scholar
  56. 56.
    Vaughan-Lee, M.: The Restricted Burnside Problem. London Mathematical Society Monographs, New Series, vol. 8, 2nd edn. Clarendon/Oxford University Press, New York (1993)Google Scholar
  57. 57.
    Völklein, H.: Groups as Galois Groups. An Introduction. Cambridge Studies in Advanced Mathematics, vol. 53. Cambridge University Press, Cambridge (1996)Google Scholar
  58. 58.
    Walsh, T.R.S.: Hypermaps versus bipartite maps. J. Comb. Theory Ser. B 18, 155–163 (1975)CrossRefzbMATHGoogle Scholar
  59. 59.
    Weil, A.: The field of definition of a variety. Am. J. Math. 78, 509–524 (1956)CrossRefzbMATHMathSciNetGoogle Scholar
  60. 60.
    Wilson, R.A.: The Finite Simple Groups. Springer, London (2009)CrossRefzbMATHGoogle Scholar
  61. 61.
    Wolfart, J.: The ‘obvious’ part of Belyi’s theorem and Riemann surfaces with many automorphisms. In: Schneps, L., Lochak, P. (eds.) Geometric Galois Actions 1. London Mathematical Society Lecture Note Series, vol. 242, pp. 97–112. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  62. 62.
    Wolfart, J.: ABC for polynomials, dessins d’enfants, and uniformization – a survey. In: Schwarz, W., Steuding, J. (eds.) Elementare und Analytische Zahlentheorie. Proceedings ELAZ-Conference, Tagungsband, 24–28 May 2004, pp. 313–343. Steiner, Stuttgart (2006). http://www.math.uni-frankfurt.de/~wolfart/

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of MathematicsUniversity of SouthamptonSouthamptonUK

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