Generating Hyperbolic Isometry Groups by Elementary Matrices
Number Theory
2023-02-13 v3
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.
Cite
@article{arxiv.2109.05054,
title = {Generating Hyperbolic Isometry Groups by Elementary Matrices},
author = {Arseniy and Sheydvasser},
journal= {arXiv preprint arXiv:2109.05054},
year = {2023}
}