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.