Some results on quotients of triangle groups

Given positive integers k, l, m , the (k, l, m) triangle group has presentation δ (k, l, m) = < X, Y, Z | X k = Y l = Z m = XYZ = 1 >. This paper considers finite permutation representations of such groups. In particular it contains descriptions of graphical and computational techniques for handling them, leading to new results on minimal two-element generation of the finite alternating and symmetric groups and the group of Rubik's cube. Applications to the theory of regular maps and automorphisms of surfaces are also discussed.