The trouble with Bianchi groups
If O is the ring of integers in a number field, PSL_n(O) is almost always generated by its elementary matrices. The only exceptions are when n=2 and O is non-Euclidean, imaginary quadratic. In these cases, group presentations are computed algorithmically. There is no known, general formulation. That these algorithms eventually terminate (and when they terminate) follows from a bound on the complexity of a Ford domain associated to PSL_2(O). We give a new upper bound on this complexity that comes within a power of log|Δ| of the value in question, where Δ is the discriminant of O.
