(2,3)-generation of exceptional groups

We study two aspects of generation of large exceptional groups of Lie type. First we show that any finite exceptional group of Lie rank at least four is (2,3)-generated, that is, a factor group of the modular group PSL2(Z). T his completes the study of (2,3)-generation of groups of Lie type. Second, we complete the proof that groups of type E-7 and E-8 over fields of odd ch aracteristic occur as Galois groups of geometric extensions of Q(ab)(t), wh ere Q(ab) denotes the maximal Abelian extension field of Q. Finally, we sho w that all finite simple exceptional groups of Lie type have a pair of stro ngly orthogonal classes. The methods of proof in all three cases are very s imilar and require the Lusztig theory of characters of reductive groups ove r finite fields as well as the classification of finite simple groups.