Beauville Surfaces and Probabilistic Group Theory

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


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 



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.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Fachbereich Mathematik und InformatikUniversität MünsterMünsterGermany

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