Arithmeticity of orbifold generalised triangle groups
type: Item
instance of: scholarly article
title: Arithmeticity of orbifold generalised triangle groups
zbMATH Open document ID: 1016.20030
DOI: 10.1016/S0022-4049(02)00180-9
author: Gerald Williams
published in: Journal of Pure and Applied Algebra
publication date: 2003-03-09T00:00:00Z
review text: \textit{C. Maclachlan} and \textit{G. J. Martin} [J. Reine Angew. Math. 511, 95-117 (1999; Zbl 0960.20029)], initiated a programme to investigate 2-generator arithmetic Kleinian groups. They proved that only finitely many arithmetic Kleinian groups can be generated by 2 elliptic elements, and they classified these groups in the non-cocompact case. In the paper under review, the authors investigate the cocompact case, restricting to a class of generalized triangle groups admitting a presentation \[ \langle a,b\mid a^l=b^m=w^n=1\rangle, \] where \(l,m,n\geq 2\) and where \(w\) is a cyclically reduced word in the free product on \(\{a,b\}\). These groups were considered before by Jones and Reid, and they arise as fundamental groups of 3-dimensional orbifolds. The authors address the question as to which generalized triangle groups admit faithful representations as cocompact arithmetic Kleinian groups. They give a table containing a list of \(l\), \(m\) and \(n\) together with conditions on the word \(w\) for which a generalized triangle group admits a faithful representation as an arithmetic Kleinian group. They also give a description of the corresponding orbifold.
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1878465
zbMATH Keywords: arithmetic groups
zbMATH Keywords: Kleinian groups
zbMATH Keywords: triangle groups
zbMATH Keywords: fundamental groups
zbMATH Keywords: orbifolds
zbMATH Keywords: cocompact groups
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
cites work: Generalized triangle groups
cites work: Q3927897
cites work: 2-Generator arithmetic Kleinian groups III
cites work: Q3802262
cites work: Two-Generator Arithmetic Kleinian Groups II
cites work: Arithmeticity, discreteness and volume
cites work: Stability and extremality in j⊘rgensen's inequality
cites work: Commutators, collars and the geometry of Möbius groups
cites work: Q4896356
cites work: Q4854333
cites work: Q3128147
cites work: Minimal index torsion-free subgroups of Kleinian groups
cites work: The non-compact arithmetic generalized triangle groups.
cites work: 2-generator arithmetic Kleinian groups
cites work: Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups
cites work: Q4028064
cites work: A note on Trace-Fields of Kleinian Groups
cites work: A characterization of arithmetic Fuchsian groups
cites work: Q4337982
cites work: Euler characteristics for orbifold generalized triangle groups
full work available at URL: s0022-4049(02)00180-9
OpenAlex ID: W2024949356
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description: scientific article; zbMATH DE number 1878465