On the discreteness criterion in \(SL (2, \mathbb{C})\)
type: Item
instance of: scholarly article
title: On the discreteness criterion in \(SL (2, \mathbb{C})\)
zbMATH Open document ID: 1213.30047
author: Shihai Yang
published in: Mathematische Zeitschrift
publication date: 2010-05-10T00:00:00Z
review text: \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.
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 5706063
zbMATH Keywords: Möbius groups
zbMATH Keywords: \(\text{SL}(2,\mathbb{C})\)
zbMATH Keywords: discrete groups
reviewed by: Jayadev S. Athreya
MaRDI profile type: Publication
full work available at URL: s00209-006-0016-0
OpenAlex ID: W1983162598
cites work: Q3310101
cites work: Inequalities in Discrete Subgroups of PSL(2, R)
cites work: Systems of generators of subgroups of PSL(2,\({\mathbb{C}})\)
cites work: On Discrete Groups of Mobius Transformations
cites work: Minimal generating systems of a subgroup of SL(2, ℂ)
cites work: Quasiconformal homeomorphisms and dynamics. II: Structural stability implies hyperbolicity for Kleinian groups
cites work: Discreteness of subgroups of $\mathrm{SL}(2,\boldsymbol{C})$ containing elliptic elements
DOI: 10.1007/S00209-006-0016-0
Recommended article: Discreteness of subgroups of \(PU(1,n;\mathbb {C})\)
Recommended article: DISCRETENESS CRITERIA FOR MÖBIUS GROUPS ACTING ON II
Recommended article: Quaternionic Möbius Transformations and Subgroups
Recommended article: Discreteness and convergence of Möbius groups
Recommended article: Elliptic elements in Möbius groups
Recommended article: Q5459664
Recommended article: MARKED SCHOTTKY GROUPS AND REPRESENTATIONS OF SURFACE GROUPS
Recommended article: On the discreteness of Möbius groups in all dimensions
Recommended article: Discreteness criteria for subgroups in SL(2, C)
Recommended article: Elliptic elements in Möbius groups.
description: scientific article; zbMATH DE number 5706063