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arXiv · 2109.05054

Generating Hyperbolic Isometry Groups by Elementary Matrices

Arseniy·Sheydvasser
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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.

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Arseniy, Sheydvasser. 2021-09-10. Generating Hyperbolic Isometry Groups by Elementary Matrices. https://arxiv.org/abs/2109.05054

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