Discreteness and convergence of Möbius groups (Q1826903)

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.