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 2 dim 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 Ta,b,c has only finitely many simple images of the form G(pr). We also obtain further results on the more general form of the conjecture, where the images G(pr) can be arbitrary quasisimple groups of type G.