On finite simple images of triangle groups

For a simple algebraic group G in characteristic p, a triple (a,b,c) of positive integers is said to be rigid for G if the dimensions of the subvarieties of G of elements of order dividing a,b,c sum to 2dim G. In this paper we complete the proof of a conjecture of the third author, that for a rigid triple (a,b,c) for G with p>0, the triangle group T_{a,b,c} has only finitely many simple images of the form G(p^r). We also obtain further results on the more general form of the conjecture, where the images G(p^r) can be arbitrary quasisimple groups of type G.