A class of 2-generator subgroups of PSL(2,\({\mathbb{C}})\)
type: Item
instance of: scholarly article
title: A class of 2-generator subgroups of PSL(2,\({\mathbb{C}})\)
zbMATH Open document ID: 0705.20047
DOI: 10.1007/BF00971262
author: E. Ya. Klimenko
published in: Siberian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Let G be the subgroup of PSL(2,\({\mathbb{C}})\) generated by the elements A and B, where A has two fixed points, and B maps one fixed point of A onto the other. The following results were obtained by \textit{B. Maskit} [Proc. Am. Math. Soc. 106, No.1, 175-186 (1989; Zbl 0676.30028)]: (i) if G is discrete, then either A is elliptic of order 2, 3, 4, or 6, or B is elliptic of order 2; (ii) if B is elliptic of order 2, then G is discrete if and only if A is a loxodromic or elliptic element of finite order; (iii) if A is elliptic of order 2, 3, 4, or 6, and B is parabolic hyperbolic or primitive elliptic then G is discrete. In the present paper the author shows that if in the case (iii) B is a non-primitive elliptic element, then the group G is not discrete. The question of discreteness of G in the case (iii) if B is loxodromic is still open.
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 4156785
zbMATH Keywords: discrete subgroup of Möbius group
zbMATH Keywords: subgroup of PSL(2,\({\mathbb{C}})\)
zbMATH Keywords: elliptic element of finite order
zbMATH Keywords: parabolic hyperbolic
zbMATH Keywords: primitive elliptic
reviewed by: Alexander Mednykh
MaRDI profile type: Publication
cites work: Q4039828
cites work: The classification of discrete 2-generator subgroups of PSL(2,R)
full work available at URL: https://doi.org/10.1007/bf00971262
OpenAlex ID: W2007444079
description: scientific article; zbMATH DE number 4156785