Advertisement

Beauville Surfaces and Probabilistic Group Theory

  • Shelly GarionEmail author
Conference paper
  • 500 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 123)

Abstract

A Beauville surface is a complex algebraic surface that can be presented as a quotient of a product of two curves by a suitable action of a finite group. Bauer, Catanese and Grunewald have been able to intrinsically characterize the groups appearing in minimal presentations of Beauville surfaces in terms of the existence of a so-called “Beauville structure”. They conjectured that all finite simple groups, except \(A_5\), admit such a structure. This conjecture has recently been proved by Guralnick-Malle and Fairbairn-Magaard-Parker. In this survey we demonstrate another approach towards the proof of this conjecture, based on probabilistic group-theoretical methods, by describing the following three works. The first is the work of Garion, Larsen and Lubotzky, showing that the above conjecture holds for almost all finite simple groups of Lie type. The second is the work of Garion and Penegini on Beauville structures of alternating groups, based on results of Liebeck and Shalev, and the third is the case of the group \({{\mathrm{PSL}}}_2(p^e)\), in which we give bounds on the probability of generating a Beauville structure. We also discuss other related problems regarding finite simple quotients of hyperbolic triangle groups and present some open questions and conjectures.

2000 Mathematics Subject Classification:

20D06 20H10 14J10 14J29 30F99 

Notes

Acknowledgments

The author would like to thank her co-organizers, Ingrid Bauer and Alina Vdovina, and all the participants in the workshop “Beauville surfaces and groups 2012” for their assistance and useful conversations, as well as the University of Newcastle for hosting the workshop. The author is grateful to the referee for pointing out further relevant references.

References

  1. 1.
    T. Bandman, S. Garion, Surjectivity and Equidistribution of the word \(x^ay^b\) on PSL \((2, q)\) and SL \((2, q)\). Int. J. Algebra Comput. 22(2) (2012)Google Scholar
  2. 2.
    I. Bauer, F. Catanese, F. Grunewald, Beauville surfaces without real structures, Geometric Methods in Algebra and Number Theory, vol. 235, Progress in Mathematics (Birkhäuser, Boston, 2005), pp. 1–42CrossRefGoogle Scholar
  3. 3.
    I. Bauer, F. Catanese, F. Grunewald, Chebycheff and Belyi polynomials, dessins d’enfants, Beauville surfaces and group theory. Mediterr. J. Math. 3(2), 121–146 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    A. Beauville, Surfaces algébriques complexes, Astérisque, 54 Paris, (1978)Google Scholar
  5. 5.
    F. Catanese, Fibred surfaces, varieties isogenous to a product and related moduli spaces. Am. J. Math. 122, 1–44 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    J. Cohen, On non-Hurwitz groups and non-congruence subgroups of the modular group. Glasg. Math. J. 22, 1–7 (1981)CrossRefzbMATHGoogle Scholar
  7. 7.
    M.D.E. Conder, Generators for alternating and symmetric groups. J. Lond. Math. Soc. 22, 75–86 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    M.D.E. Conder, Hurwitz groups: a brief survey. Bull. Am. Math. Soc. 23, 359–370 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    M.D.E. Conder, An update on Hurwitz groups. Groups, Complex. Cryptol. 2, 35–40 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    L. Di Martino, M.C. Tamburini, A.E. Zalesski, On Hurwitz groups of low rank. Commun. Algebra 28(11), 5383–5404 (2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    L.E. Dickson, Linear Groups with an Exposition of the Galois Field Theory (Teubner, Leipzig, 1901)zbMATHGoogle Scholar
  12. 12.
    J.D. Dixon, The probability of generating the symmetric group. Math. Z. 110, 199–205 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    B. Everitt, Alternating quotients of Fuchsian groups. J. Algebra 223, 457–476 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    B. Fairbairn, K. Magaard, C. Parker, Generation of finite simple groups with an application to groups acting on Beauville surfaces, to appear in Proc. Lond. Math. SocGoogle Scholar
  15. 15.
    D. Frohardt, K. Magaard, About a conjecture of Guralnick and Thompson, in Groups, Difference Sets, and the Monster, Proceedings of the Ohio State Conference on Groups and Geometrie, ed. by K.T. Arasu, et al. (Walter de Gruyter, Berlin, 1996), pp. 43–54Google Scholar
  16. 16.
    Y. Fuertes, G. González-Diez, On Beauville structures on the groups \(S_n\) and \(A_n\). Math. Z. 264, 959–968 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Y. Fuertes, G. Jones, Beauville surfaces and finite groups. J. Algebra 340, 13–27 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    S. Garion, On Beauville structures for PSL \((2, q)\). arXiv:1003.2792
  19. 19.
    S. Garion, M. Larsen, A. Lubotzky, Beauville surfaces and finite simple groups. J. Reine Angew. Math. 666, 225–243 (2012)zbMATHMathSciNetGoogle Scholar
  20. 20.
    S. Garion, M. Penegini, New Beauville surfaces and finite simple groups. Manuscripta Math. 142, 391–408 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    D. Gluck, K. Magaard, Character and fixed point ratios in finite classical groups. Proc. Lond. Math. Soc. 71, 547–584 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    R. Gow, Commutators in finite simple groups of Lie type. Bull. Lond. Math. Soc. 32(3), 311–315 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    R. Guralnick, J. Thompson, Finite groups of genus zero. J. Algebra 131, 303–341 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    R. Guralnick, C. Praeger, T. Penttila, J. Saxl, Linear groups with orders having certain large prime divisors. Proc. Lond. Math. Soc. 78, 167–214 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    R. Guralnick, G. Malle, Simple groups admit Beauville structures. J. Lond. Math. Soc. 85(3), 694–721 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    W.M. Kantor, A. Lubotzky, The probability of generating a finite classical group. Geom. Dedicata 36, 67–87 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    V. Landazuri, G.M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups. J. Algebra 32, 418–443 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    U. Langer, G. Rosenberger, Erzeugende endlicher projektiver linearer Gruppen. Results Math. 15(1–2), 119–148 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    M. Larsen, A. Lubotzky, C. Marion, Deformation theory and finite simple quotients of triangle groups I. arXiv:1301.2949
  30. 30.
    F. Levin, G. Rosenberger, Generators of finite projective linear groups, II. Results Math. 17(1–2), 120–127 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    M.W. Liebeck, A. Shalev, The probability of generating a finite simple group. Geom. Dedicata 56, 103–113 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    M.W. Liebeck, A. Shalev, Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks. J. Algebra 276, 552–601 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    M.W. Liebeck, A. Shalev, Fuchsian groups, finite simple groups and representation varieties. Invent. Math. 159(2), 317–367 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    M.W. Liebeck, A. Shalev, Character degrees and random walks in finite groups of Lie type. Proc. Lond. Math. Soc. (3) 90(1), 61–86 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    M.W. Liebeck, B.M.S. Martin, A. Shalev, On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function. Duke Math. J. 128(3), 541–557 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    A. Lucchini, M.C. Tamburini, Classical groups of large rank as Hurwitz groups. J. Algebra 219, 531–546 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    A. Lucchini, M.C. Tamburini, J.S. Wilson, Hurwitz groups of large rank. J. Lond. Math. Soc. 61, 81–92 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    A.M. Macbeath, Generators of the linear fractional groups, Number Theory. Proc. Sympos. Pure Math. XII, Houston, Tex. (1967), Am. Math. Soc., Providence, R.I. (1969), 14–32Google Scholar
  39. 39.
    G. Malle, Hurwitz groups and \(G_2(q)\). Can. Math. Bull. 33, 349–357 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    C. Marion, Triangle groups and PSL \(_2(q)\). J. Group Theory 12, 689–708 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    C. Marion, On triangle generation of finite groups of Lie type. J. Group Theory 13, 619–648 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    C. Marion, Triangle generation of finite exceptional groups of low rank. J. Algebra 332, 35–61 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    C. Marion, Random and deterministic triangle generation of three-dimensional classical groups I. Commun. Algebra 41(3), 797–852 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    M. Suzuki, Group Theory I (Springer, Berlin, 1982)CrossRefzbMATHGoogle Scholar
  45. 45.
    M.C. Tamburini, A.E. Zalesski, Classical groups in dimension \(5\) which are Hurwitz, Finite groups 2003 (Walter de Gruyter, Berlin, 2004)Google Scholar
  46. 46.
    M.C. Tamburini, M. Vsemirnov, Hurwitz groups and Hurwitz generation, in Handbook of Algebra, vol. 4, ed. by M. Hazewinkel (Elsevier, Amsterdam, 2006), pp. 385–426Google Scholar
  47. 47.
    M.C. Tamburini, M. Vsemirnov, Irreducible \((2,3,7)\)-subgroups of PGL \(_n(F)\) for \(n\le 7\). J. Algebra 300(1), 339–362 (2006)Google Scholar
  48. 48.
    R. Vincent, A.E. Zalesski, Non-Hurwitz classical groups. LMS J. Comput. Math. 10, 21–82 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  49. 49.
    R.A. Wilson, The Monster is a Hurwitz group. J. Group Theory 4(4), 367–374 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  50. 50.
    E. Witten, On quantum gauge theories in two dimensions. Commun. Math. Phys. 141, 153–209 (1991)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Fachbereich Mathematik und InformatikUniversität MünsterMünsterGermany

Personalised recommendations