Determining finite simple images of finitely presented groups

I will discuss joint work with Martin Bridson and Martin Liebeck which addresses the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative results. For a collection of finite simple groups that contains infinitely many alternating groups, or contains classical groups of unbounded dimensions, we prove that there is no such algorithm. On the other hand, for a collection of simple groups of fixed Lie type we obtain positive results by using the model theory of finite fields.