Simple groups admit Beauville structures. (Q2890307)

According to \textit{F. Catanese} [Am. J. Math. 122, No. 1, 1-44 (2000; Zbl 0983.14013)], a Beauville surface is a compact complex surface \(S\) which is rigid (that is, it has no non-trivial deformations) and satisfies \(S=(X\times Y)/G\), where \(X\) and \(Y\) are curves of genus at least 2 and \(G\) is a finite group acting freely on \(X\times Y\). A finite group \(G\) is said to admit an unmixed Beauville structure if there exist two pairs of generators \((x_1,y_1)\) and \((x_2,y_2)\) for \(G\) such that \(\Sigma(x_1,y_1)\cap\Sigma(x_2,y_2)=\{1\}\), where, for \(x,y\in G\), \(\Sigma(x,y)\) is the union of conjugacy classes of all powers of \(x\), all powers of \(y\), and all powers of \(xy\). \textit{I. Bauer, F. Catanese} and \textit{F. Grunewald} [Mediterr. J. Math. 3, 121-146 (2006; Zbl 1167.14300)] conjectured that all finite non-Abelian simple groups other than the alternating group of degree 5 admit unmixed Beauville structures.NEWLINENEWLINE The first main result of the present paper is a proof of this conjecture. A proof of the conjecture was given by \textit{S. Garion, M. Larsen} and \textit{A. Lubotzky} [J. Reine Angew. Math. 666, 225-243 (2012; Zbl 1255.20008)] for all sufficiently large non-Abelian simple groups, by \textit{Y. Fuertes} and \textit{G. González-Diez} [Math. Z. 264, No. 4, 959-968 (2010; Zbl 1190.30027)] for the alternating groups, \textit{Y. Fuertes} and \textit{G. A. Jones} [J. Algebra 340, No. 1, 13-27 (2011; Zbl 1233.14027)] and by \textit{S. Garion} and \textit{N. Penegini} [``New Beauville surfaces and finite simple groups'', \url{arXiv:0910.5402}] for some low rank cases. The authors note that \textit{B. Fairbain, K. Magaard} and \textit{C. Parker} [``Generation of finite simple groups with an application to groups acting on Beauville surfaces'', \url{arXiv:1010.3500}] also proved the conjecture using similar methods.NEWLINENEWLINE The second main result of the present paper is a proof of an analog of the first result for simple algebraic groups which depends on some upper bounds for character values of regular semisimple elements in finite groups of Lie type. Finally, it is proved that any finite simple group contains two conjugacy classes \(C\) and \(D\) such that any pair of elements in \(C\times D\) generates the group. The authors note that \textit{W. M. Kantor, A. Lubotzky} and \textit{A. Shalev} [J. Algebra 348, No. 1, 302-314 (2011; Zbl 1248.20036)] proved a slighly weaker version of the last result.