Generating Hyperbolic Isometry Groups by Elementary Matrices

2021/09/10 by Arseniy, Sheydvasser
#11A05 #20G30 #22E40 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2109.05054

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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