Beauville surfaces and finite simple groups. (Q2890366)

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 In the present paper, the authors prove this is so for almost all finite simple groups, i.e., with at most finitely many exceptions. The proof makes use the structure theory of finite simple groups, probability theory, and character estimates. The Bauer-Catanese-Grunewald conjecture has now been proved by \textit{R. Guralnick} and \textit{G. Malle} [J. Lond. Math. Soc., II. Ser. 85, No. 3, 694-721 (2012; Zbl 1255.20009)] and \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}].