Computing in matrix groups over infinite domains is a relatively new area of computational group theory. As a consequence of the `Tits alternative', most of the activity in this area has concentrated on the class of solvable-by-finite groups. In this talk we describe our development of efficient algorithms for computing with nilpotent matrix groups over finite and infinite fields. These algorithms deal with standard computational group theory problems such as nilpotency testing, constructing presentations, and deciding finiteness. The main technique is based on change of the ground domain via congruence homomorphism. We also rely on other methods and results of linear group theory, especially structural results for nilpotent linear groups.
This is joint work with A. S. Detinko (NUI, Galway).