From the Poincaré theorem to generators of the unit group of integral group rings of finite groups

We give an algorithm to determine finitely many generators for a subgroup of finite index
in the unit group of an integral group ring ZG of a finite nilpotent group G, this provided
the rational group algebra QG does not have simple components that are division classical
quaternion algebras or two-by-two matrices over a classical quaternion algebra with centre
Q. The main difficulty is to deal with orders in quaternion algebras over the rationals or a
quadratic imaginary extension of the rationals. In order to deal with these we give a finite and
easy implementable algorithm to compute a fundamental domain in the hyperbolic three space
H 3 (respectively hyperbolic two space H 2 ) for a discrete subgroup of PSL 2 (C) (respectively
PSL 2 (R)) of finite covolume. Our results on group rings are a continuation of earlier work of
Ritter and Sehgal, Jespers and Leal.