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.


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