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Jørgensen’s inequality for discrete convergence groups. (English) Zbl 0945.30035
We explore in this paper whether certain fundamental properties of the action of Kleinian groups on the Riemann sphere extend to the action of discrete convergence groups on $$\overline{\mathbf R^2}$$. A Jørgensen inequality for discrete $$K$$-quasiconformal groups is developed, and it is shown that such an inequality depends naturally on the quasiconformal dilatation $$K$$. Furthermore, it is established that no such inequality can hold for general discrete convergence groups. In the discontinuous case a universal constraint on discreteness is formulated for both quasiconformal and general convergence groups.

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 57S30 Discontinuous groups of transformations 30C62 Quasiconformal mappings in the complex plane 20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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