Free products of units in algebras. I: Quaternion algebras
type: Item
instance of: scholarly article
title: Free products of units in algebras. I: Quaternion algebras
zbMATH Open document ID: 0934.16026
author: Arnaldo Mandel
author: J. Z. Gonçalves
author: M. Shirvani
published in: Journal of Algebra
publication date: 2000-01-27T00:00:00Z
review text: It is well known that a non-Abelian division ring \(D\) which is finite dimensional over its center contains a non-Abelian free subgroup. This implies, of course, that a typical pair of elements of \(D\) generates a free subgroup of rank two. Nevertheless, it is not easy to exhibit a free pair of elements. The authors are looking for the pairs of elements which are either free or ``nearly free'' in quaternion algebras over domains. To be more specific they call two elements semifree if they generate the free product of the corresponding cyclic subgroups in the ambient group of units. They also introduce the notion of being semifree modulo a normal subgroup. They suggest sufficient conditions, which are expressed in terms of valuations, for a pair of elements to be either free or semifree modulo the center. They apply these results to exhibit free subgroups of the group \(\text{SO}(3,\mathbb{R})\) of real orthogonal matrices reproving and generalizing previously known results of this kind. They also exhibit free pairs in the Malcev-Neumann skewfields generated by residually torsion-free nilpotent groups and give some further applications.
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 16S36
zbMATH DE Number: 1275265
zbMATH Keywords: free groups
zbMATH Keywords: groups of units
zbMATH Keywords: quaternion algebras
zbMATH Keywords: skew polynomial rings
zbMATH Keywords: free pairs of elements
zbMATH Keywords: semifree pairs
zbMATH Keywords: free products
zbMATH Keywords: Malcev-Neumann skewfields
reviewed by: Leonid Makar-Limanov
MaRDI profile type: Publication
full work available at URL: jabr.1998.7680
OpenAlex ID: W2030743146
cites work: On faithful representations of free products of groups
cites work: Q5661659
cites work: On Free Products of Cyclic Rotation Groups
cites work: Q3963168
cites work: On Embedding of Group Rings of Residually Torsion Free Nilpotent Groups into Skew Fields
cites work: Free Groups in Subnormal Subgroups and the Residual Nilpotence of the Group of Units of Groups Rings
cites work: Free group algebras in certain division rings
cites work: Are there free groups in division rings?
cites work: Free group algebras in the field of fractions of differential polynomial rings and enveloping algebras
cites work: Orthogonal Isomorphic Representations Of Free Groups
cites work: Free Subgroups in the Unit Groups of Integral Group Rings
cites work: On Subgroups of the Multiplicative Group of Skew Fields
cites work: Free Subgroups of Normal Subgroups of the Multiplicative Group of Skew Fields
cites work: On normal subgroups of multiplicative group of skew fields generated by a polycyclic-by-finite group
cites work: Valuation methods in division rings
cites work: Q4761406
cites work: Q4145882
cites work: Q3325876
cites work: On group rings of nilpotent groups
cites work: On subnormal subgroups of skew fields
cites work: Q4145843
cites work: A class of free rotation groups
cites work: Free subgroups in linear groups
DOI: 10.1006/JABR.1998.7680
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description: scientific article; zbMATH DE number 1275265