## 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

### MSC:

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