Generating hyperbolic isometry groups by elementary matrices

Abstract

We consider three families of groups: the Bianchi groups where is the ring of integers of an imaginary, quadratic field; the groups where is a -order of a definite, rational quaternion algebra with an orthogonal involution; and the groups where is an order of a definite, rational quaternion algebra. We show that such groups are generated by elementary matrices if and only if is semi-Euclidean (or -semi-Euclidean), which is a generalization of the usual notion of a Euclidean ring. The proofs are surprisingly simple and proceed by considering fundamental domains of Kleinian groups.