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

Abstract

We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring $\mathbb{Z} G$ of a finite nilpotent group $G$, this provided the rational group algebra $\mathbb{Q} G$ does not have simple components that are division classical quaternion algebras or two-by-two matrices over a classical quaternion algebra with centre $\mathbb{Q}$. 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 $\mathbb{H}^3$ (respectively hyperbolic two space $\mathbb{H}^2$) for a discrete subgroup of ${\rm PSL}_2(\mathbb{C})$ (respectively ${\rm PSL}_2(\mathbb{R})$) of finite covolume. Our results on group rings are a continuation of earlier work of Ritter and Sehgal, Jespers and Leal.