The bounded model for hyperbolic 3-space and a quaternionic uniformization theorem. (English) Zbl 0555.30031

The author gives a succinct method of describing the elements of the directly conformal group of \(\Delta^ 3\), the bounded model of \({\mathcal H}^ 3\), hyperbolic 3-space, by utilising invertible \(2\times 2\) matrices over the quaternions. The Ford region for the action of a general discrete group of orientation-preserving isometries of \({\mathcal H}^ 3\) can thus conveniently be discussed and the author proves a Poincaré estimate for such groups in terms of the usual \(SL_ 2({\mathbb{C}})\) representation and extends known results in this direction. This model also permits a suitable definition of Poincaré \(\theta\)- series and leads to a uniformization of hyperbolic 3-manifolds \(\Delta^ 3/G\) in \(\hat H^ 2\), where \(\hat H\) denotes the extended quaternions.
Reviewer: C.Maclachlan


30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20H15 Other geometric groups, including crystallographic groups
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