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

We generalize an algorithm established in earlier work [21] to compute finitely
many generators for a subgroup of finite index of an arithmetic group acting properly discon-
tinuously 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
of finite index subgroups of a discrete subgroup of Vahlen’s group, i.e. a group of 2-by-2
matrices with entries in the Clifford algebra satisfying certain conditions. The motivation
comes from units in integral group rings and this new algorithm allows to handle unit groups
of orders in 2-by-2 matrices over rational quaternion algebras. The rings investigated are
part of the so-called exceptional components of a rational group algebra.