The finite subgroups of maximal arithmetic Kleinian groups
type: Item
instance of: scholarly article
title: The finite subgroups of maximal arithmetic Kleinian groups
zbMATH Open document ID: 0973.20040
author: Ted Chinburg
author: Eduardo Friedman
publication date: 2000-12-14T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75471
review text: The finite subgroups of \(\text{PGL}(2,\mathbb{C})\) have been known since Klein's time to be isomorphic to a cyclic group, a dihedral group, \(A_4\), \(S_4\) or \(A_5\). It is useful to know which of these finite groups appear as subgroups of a given Kleinian group \(\Gamma\subset\text{PGL}(2,\mathbb{C})\). The authors treat here the more restricted problem of computing the finite subgroups of a Kleinian group corresponding to a minimal arithmetic hyperbolic 3-orbifold, where minimal means that the orbifold does not properly cover any other orbifold. This subject is linked to interesting aspects of indefinite quaternion algebras over a number field and of an assignment, linked to dihedral subgroups of arithmetic Kleinian groups, of an ideal class to a pair of global Hilbert symbols. They give a lot of interesting primers. Note that while their main interest is in finite subgroups of a maximal arithmetic Kleinian group \(\Gamma\subset\text{PGL}(2,\mathbb{C})\), the authors actually deal throughout with the more general case of a maximal irreducible arithmetic group \(\Gamma\subset\text{PGL}(2,\mathbb{R})^n\times\text{PGL}(2,\mathbb{C})^m\).
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 11F06
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1544081
zbMATH Keywords: finite subgroups
zbMATH Keywords: Kleinian groups
zbMATH Keywords: minimal arithmetic hyperbolic 3-orbifolds
zbMATH Keywords: quaternion algebras over number fields
reviewed by: V. V. Chueshev
MaRDI profile type: Publication
OpenAlex ID: W2319805123
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DOI: 10.5802/AIF.1807
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 1544081