Abstract
In this paper we construct new Beauville surfaces with group either PSL(2, p e), or belonging to some other families of finite simple groups of Lie type of low Lie rank, or an alternating group, or a symmetric group, proving a conjecture of Bauer, Catanese and Grunewald. The proofs rely on probabilistic group theoretical results of Liebeck and Shalev, on classical results of Macbeath and on recent results of Marion.
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Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact complex surfaces. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4. Springer, Berlin (2004)
Bauer, I., Catanese, F., Grunewald, F.: Beauville surfaces without real structures. In: Geometric Methods in Algebra and Number Theory, Progress in Mathematics, vol. 235, pp. 1–42, Birkhäuser, Boston (2005)
Bauer I., Catanese F., Grunewald F.: Chebycheff and Belyi polynomials, dessins d’enfants, Beauville surfaces and group theory. Mediterr. J. Math. 3(2), 121–146 (2006)
Beauville, A.: Surfaces Algébriques Complex. Astérisque, vol. 54. Paris (1978)
Catanese F.: Fibred surfaces, varieties isogenous to a product and related moduli spaces. Am. J. Math. 122, 1–44 (2000)
Conder M.D.E.: Generators for alternating and symmetric groups. J. Lond. Math. Soc. 22, 75–86 (1980)
Conder M.D.E.: Hurwitz groups: a brief survey. Bull. Am. Math. Soc. 23, 359–370 (1990)
Dickson, L.E.: Linear Groups with an Exposition of the Galois Field Theory (Teubner 1901).
Everitt B.: Alternating quotients of Fuchsian groups. J. Algeb. 223, 457–476 (2000)
Fuertes Y., Jones G.: Beauville surfaces and finite groups. J. Algeb. 340, 13–27 (2011)
Garion, S.: On Beauville structures for PSL(2, q). Preprint availiable at arXiv:1003.2792
Garion S., Larsen M., Lubotzky A.: Beauville surfaces and finite simple groups. J. Reine Angew. Math. 666, 225–243 (2012)
Gorenstein D.: Finite groups. Chelsea, New York (1980)
Liebeck M.W., Shalev A.: Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks. J. Algeb. 276, 552–601 (2004)
Liebeck M.W., Shalev A.: Fuchsian groups, finite simple groups and representation varieties. Invent. Math. 159(2), 317–367 (2005)
Macbeath, A.M.: Generators of the linear fractional groups. In: Number Theory (Proceedings of Symposia in Pure Mathematics, vol. XII, Houston 1967), pp. 14–32. AMS, Providence (1969)
Marion C.: Triangle groups and PSL2(q), J. Group Theory 12, 689–708 (2009)
Marion C.: Triangle generation of finite exceptional groups of low rank. J. Algeb. 332(1), 35–61 (2011)
Marion C.: Random and deterministic triangle generation of three-dimentional classical groups I. Preprint.
Marston M.D.E.: An update on Hurwitz groups. Group. Complex. Cryptol. 2(1), 35–49 (2010)
Serrano, F.: Isotrivial fibred surfaces. Annali di Matematica. pura e applicata CLXXI, 63–81 (1996)
Suzuki M.: Group Theory I. Springer, Berlin (1982)
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Garion, S., Penegini, M. New Beauville surfaces and finite simple groups. manuscripta math. 142, 391–408 (2013). https://doi.org/10.1007/s00229-013-0607-0
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DOI: https://doi.org/10.1007/s00229-013-0607-0