On the discreteness criterion in \(SL (2, \mathbb{C})\) (Q970195)

\textit{T. Jørgensen} [Am. J. Math. 98, 739-749 (1976; Zbl 0336.30007)] proved that a non-elementary subgroup \(G\) of Möbius transformations acting on \(\mathbb{S}^2\) is discrete if and only if, for each \(f\) and \(g\) in \(G\), the group \(\langle f,g\rangle\) is discrete. A number of authors has shown that, under special conditions, the pairs \((f,g)\) that must be considered can be restricted, say to hyperbolic, loxodromic, or elliptic elements. The author recounts the precise history and adds three additional results to the list: A non-elementary subgroup \(G\) of \(\text{SL}(2,\mathbb{C})\) is discrete if (i) \(G\) contains parabolics and elliptics, and each \(\langle f,g\rangle\) is discrete for each \(f\) parabolic and each \(g\) elliptic; or (ii) \(G\) contains loxodromic and parabolic elements or it contains loxodromic and elliptic elements, and each \(\langle f,g\rangle\) is discrete for each \(f\) loxodromic and each \(g\) parabolic or elliptic, respectively; or (iii) \(G\) contains parabolics and \(\langle f,g\rangle\) is discrete for each \(f\) and \(g\) parabolic.