Jørgensen’s inequality for discrete convergence groups. (English) Zbl 0945.30035

Bonfert-Taylor, Petra

Ann. Acad. Sci. Fenn., Math. 25, No. 1, 131-150 (2000).

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)

Keywords:

convergence groups; quasiconformal groups; Kleinian groups; Jørgensen’s inequality