2013/10/31 by Shelly Garion, Garion, Shelly
Mathematics · #14J10 #14J29 #20D06 #20H10 #30F99 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #math.AG #math.GR #math.RT #msc:14J10 #msc:14J29 #msc:20D06 #msc:20H10 #msc:30F99
paper · pdf · doi:10.48550/arxiv.1310.8587
13 pages, a survey for the "Beauville surfaces and groups 2012" conference proceedings
arxiv created 2013/10/31 · openalex publication_date 2013/10/31 · arxiv updated 2013/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 A5, 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 PSL(2,pe), 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.