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We give an algorithm to determine finitely many generators for a subgroup offinite index in the unit group of an integral group ring $\mathbb{Z} G$ of afinite nilpotent group $G$, this provided the rational group algebra$\mathbb{Q} G$ does not have simple components that are division classicalquaternion algebras or two-by-two matrices over a classical quaternion algebrawith centre $\mathbb{Q}$. The main difficulty is to deal with orders inquaternion algebras over the rationals or a quadratic imaginary extension ofthe rationals. In order to deal with these we give a finite and easyimplementable algorithm to compute a fundamental domain in the hyperbolic threespace $\mathbb{H}^3$ (respectively hyperbolic two space $\mathbb{H}^2$) for adiscrete subgroup of ${\rm PSL}_2(\mathbb{C})$ (respectively ${\rmPSL}_2(\mathbb{R})$) of finite covolume. Our results on group rings are acontinuation of earlier work of Ritter and Sehgal, Jespers and Leal.

From the Poincar\'e Theorem to generators of the unit group of integral group rings of finite groups