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Discreteness criterion in $\mathrm{SL}(2,\mathbb{C})$ by a test map

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  • Mathematics


Cao, W. Osaka J. Math. 49 (2012), 901–907 DISCRETENESS CRITERION IN SL(2, C) BY A TEST MAP WENSHENG CAO (Received October 27, 2010, revised February 17, 2011) Abstract In the paper [12], Yang conjectured that a nonelementary subgroup G of SL(2,C) containing elliptic elements is discrete if for each elliptic element g 2 G the group h f, gi is discrete, where f 2 SL(2, C) is a test map being loxodromic or elliptic. By embedding SL(2, C) into U(1, 1IH), we give an affirmative answer to this ques- tion. As an application, we show that a nonelementary and nondiscrete subgroup of Isom(H 3) must contain an elliptic element of order at least 3. 1. Introduction The discreteness of Möbius groups is a fundamental problem, which has been dis- cussed by many authors. In 1976, Jørgensen established the following discreteness cri- terion by using the well-known Jørgensen’s inequality [8]. Theorem J. A nonelementary subgroup G of Möbius transformations acting on OC is discrete if and only if for each pair of elements f, g 2 G, the group h f, gi is discrete. This result shows that the discreteness of a nonelementary Möbius group depends on the information of all its rank two subgroups. The above result has been generalized by many authors by using information of partial rank two subgroups. For example, Gilman [5] and Isochenko [7] used each pair of loxodromic elements, Tukia and Wang [10] used each pair of elliptic elements. Sullivan [9] showed that a nonelementary and non-discrete subgroup is either dense in SL(2,C) or conjugate to a dense subgroup of SL(2,R). This result gives an approach to studying the discreteness of Möbius groups from the topological aspect. Mainly us- ing Sullivan’s result, Yang [11] obtained some generalizations by the information of the remaining four kinds of rank two subgroups. Recently, Chen [3] proposed to use a fixed Möbius transformation as a test map to test the discreteness of a given Möbius group. His result suggests that the discrete- ness is not a totally

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