  We develop theorems which produce a multitude of hyperbolic triples for the
finite classical groups. We apply these theorems to prove that every
quasisimple group except Alt(5) and SL_2(5) is a Beauville group. In
particular, we settle a conjecture of Bauer, Catanese and Grunewald which
asserts that all non-abelian finite simple groups except for the alternating
group $\Alt(5)$ are Beauville groups.
