Discreteness of subgroups of \(PU(1,n;\mathbb {C})\)
type: Item
instance of: scholarly article
title: Discreteness of subgroups of \(PU(1,n;\mathbb {C})\)
zbMATH Open document ID: 1167.30024
author: Hua Wang
author: Yue-Ping Jiang
author: Baohua Xie
published in: Proceedings of the Japan Academy. Series A
publication date: 2008-11-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1212500760
review text: 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)].
reviewed by: James W. Cannon
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 5362015
zbMATH Keywords: Kleinian group
zbMATH Keywords: \(PU(1,n
zbMATH Keywords: \mathbb{C})\)
zbMATH Keywords: elliptic transformation
zbMATH Keywords: discrete group
zbMATH Keywords: dense group
zbMATH Keywords: regular elliptic elements
MaRDI profile type: Publication
OpenAlex ID: W2082132848
cites work: Nondiscrete Groups of Hyperbolic Motions
cites work: Inequalities in Discrete Subgroups of PSL(2, R)
cites work: Q4237093
cites work: ALGEBRAIC CONVERGENCE OF FINITELY GENERATED KLEINIAN GROUPS
cites work: Notes on elements of U(1,n;\({\mathbb{C}})\)
cites work: On Discrete Subgroups of PU(1,2;C ) with Heisenberg Translations
cites work: A discrete criterion in \(\text{PU}(2,1)\) by use of elliptic elements.
DOI: 10.3792/PJAA.84.78
description: scientific article; zbMATH DE number 5362015