Generation of finite quasisimple groups with an application to groups acting on Beauville surfaces. (Q2856115)

A group \(G\) is called a Beauville group if there exists a pair of triples \((x_1,y_1,z_1),(x_2,y_2,z_2)\in G\times G\times G\) such that for \(i=1,2\) the following hold: (i) \(G=\langle x_i,y_i,z_i\rangle\) and \(x_iy_iz_i=1\); (ii) \(1/|x_i|+1/|y_i|+1/|z_i|<1\); and (iii) no non-identity power of \(x_1\), \(y_1\) or \(z_1\) is conjugate in \(G\) to a power of \(x_2\), \(y_2\) or \(z_2\). A triple \((x_1,y_1,z_1)\) of elements of a group \(G\) which satisfies the conditions (i) and (ii) is called a hyperbolic triple for \(G\). A finite Beauville group is associated with a Beauville surface of unmixed type (see the paper of \textit{F. Catanese} [Am. J. Math. 122, No. 1, 1-44 (2000; Zbl 0983.14013)]).NEWLINENEWLINE \textit{I. Bauer, F. Catanese} and \textit{F. Grunewald} [Mediterr. J. Math. 3, No. 2, 121-146 (2006; Zbl 1167.14300)] conjectured that all finite non-Abelian simple groups except for the alternating group \(\text{Alt}(5)\) are Beauville groups. In the present paper, the authors prove theorems which produce a multitude of hyperbolic triples for the finite classical groups. They apply these theorems to prove that every finite quasisimple group with the exceptions of \(\text{Alt}(5)\) and \(\text{SL}_2(5)\) is a Beauville group. In particular, they settle the Bauer-Catanese-Grunewald conjecture.NEWLINENEWLINE Note that \textit{S. Garion, M. Larsen} and \textit{A. Lubotzky} [J. Reine Angew. Math. 666, 225-243 (2012; Zbl 1255.20008)] proved the conjecture for sufficiently large groups and \textit{R. Guralnick} and \textit{G. Malle} [J. Lond. Math. Soc., II. Ser. 85, No. 3, 694-721 (2012; Zbl 1255.20009)] confirmed the conjecture simultaneously with the authors.