Discreteness of subgroups of \(PU(1,n;\mathbb {C})\)
This paper considers further generalizations, to subgroups of \(PU(1, n;\mathbb{C})\), of the following suggestive theorem of Troels Jørgensen [\textit{T. Jørgensen}, Am. J. Math. 98, 739--749 (1976; Zbl 0336.30007) and \textit{T. Jørgensen and P. Klein}, Q. J. Math., Oxf. II. Ser. 33, 325--332 (1982; Zbl 0499.30033)]: A nonelementary subgroup \(G\) of \(SL(2, \mathbb{C})\) is discontinuous if and only if all of its two-generator subgroups are discontinuous. The authors consider \(n\)-dimensional subgroups \(G\) of \(PU(1,n;\mathbb{C})\) containing elliptic elements and show that \(G\) is discrete, if and only if (1) for each pair \(f,g\in G\) with \(f\) regular elliptic and \(g\) loxodromic, \(\langle f,g\rangle\) is discrete, if and only if (2) for each pair \(f,g\in G\) loxodromic, \(\langle f,g\rangle\) is discrete, if and only if (3) for each pair \(f,g\in G\) regular elliptic, \(\langle f,g\rangle\) is discrete. For further generalizations to \(PU (1,n; \mathbb{C})\) see references in this paper and in [\textit{Wensheng Cao}, Bull. Aust. Math. Soc. 78, No. 2, 211--224 (2008; Zbl 1167.30023)]. Jørgensen's result was generalized to the \(n\)-dimension Möbius group by \textit{G. J. Martin} [Acta Math. 163, No. 3--4, 253--289 (1989; Zbl 0698.20037)] and by \textit{W. Abikoff} and \textit{A. Hass} [Bull. Lond. Math. Soc. 22, 233--238 (1990; Zbl 0709.20026)].
scientific article; zbMATH DE number 5362015
scientific article