Generating Hyperbolic Isometry Groups by Elementary Matrices

TL;DR

This paper characterizes when certain hyperbolic isometry groups, including Bianchi and quaternionic groups, are generated by elementary matrices, linking this property to the semi-Euclidean nature of their rings.

Contribution

It establishes a precise criterion connecting the generation of these groups by elementary matrices to the semi-Euclidean property of their rings, generalizing Euclidean rings.

Findings

Groups are generated by elementary matrices iff the ring is semi-Euclidean.

The proofs involve analyzing fundamental domains of Kleinian groups.

The results extend classical Euclidean ring concepts to more complex algebraic structures.

Abstract

We consider three families of groups: the Bianchi groups SL(2,O) where O is the ring of integers of an imaginary, quadratic field; the groups SL*(2,O) where O is a *-order of a definite, rational quaternion algebra with an orthogonal involution; and the groups SL(2,O) where O is an order of a definite, rational quaternion algebra. We show that such groups are generated by elementary matrices if and only if O 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.