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 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 PSL2(C) (respectively PSL2(R)) of finite covolume. Our results on group rings are a continuation of earlier work of Ritter and Sehgal, Jespers and Leal.

On units in orders in 2-by-2 matrices over quaternion algebras with rational center

“…However both algorithms work perfectly fine for discrete subgroups of PSL + (Γ n (R)), hence for Z-orders in PSL + (Γ n (R)). To keep the symmetry with the earlier work [19] and also with classical work on Fuchsian and Kleinian groups, we decided to state it this way. However in Section 6, we will need an algorithm for discrete subgroups of PSL + (Γ n (R)), which are not contained in PSL + (Γ n (Z)).…”

“…The goal of this paper is to develop a new method that deals with these components. This work is in fact a continuation of the work done in [19]. As explained above, in that paper, the authors use the action on hyperbolic space of dimension 2 and 3 and develop an algorithm, which takes as an input a discrete cofinite subgroup of PSL 2 (R) or PSL 2 (C) to return a finite-sided convex polyhedron that contains a Dirichlet fundamental domain of the discrete subgroup.…”

“…This constituted the first concrete example of an attack on the exceptional components of type 4. This was then finally generalized in [19], where an algorithm, called the DAFC, is given to compute an 'approximate' fundamental domain, and hence generators, for subgroups of finite index in groups that are contained in the unit group of orders in quaternion algebras over Q or quadratic imaginary extensions of Q, in particular for Bianchi groups. This work handled the exceptional components of type 1 and 2, as well as a part of exceptional components of type 4.…”

“…

We generalize an algorithm established in earlier work [19] to compute finitely many generators for a subgroup of finite index of a group acting discontinuously on hyperbolic space of dimension 2 and 3, to hyperbolic space of higher dimensions using Clifford algebras. We hence get an algorithm which gives a finite set of generators up to finite index of a discrete subgroup of Vahlen's group, i.e.

…”

Traces of Torsion Units

“…This has been done in only some exceptional cases (see [10,12,18]). An apparently easier problem is to study the units of finite order.…”