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Discreteness and convergence of Möbius groups. (English) Zbl 1053.30038

In this paper, the author shows that a fixed nontrivial Möbius transformation can be used as a test map to test the discreteness of a nonelementary Möbius group. Following Jorgensen’s well-known result about the discreteness of a group of Möbius transformations, the author proves two discreteness theorems. The first of these two results says that if for every element g of an n-dimensional subgroup \(G\) of I\(som(Hn)\), the group \(<g,h>\) is discrete , then \(G\) is discrete. The second discreteness theorem states some conditions for the discreteness of G. Also two convergence theorems are given at the end.

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
28A78 Hausdorff and packing measures
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