The soluble subgroups and the Tits alternative in linear groups over rings of fractions of polycyclic group rings. I
type: Item
instance of: scholarly article
title: The soluble subgroups and the Tits alternative in linear groups over rings of fractions of polycyclic group rings. I
zbMATH Open document ID: 0784.20025
DOI: 10.1016/0022-4049(93)90105-3
author: A. I. Lichtman
published in: Journal of Pure and Applied Algebra
publication date: 1993-08-08T00:00:00Z
review text: In this tour-de-force the author studies subrings of the matrix ring \(D^{n\times n}\) and subgroups of \(GL(n,D)\) for a certain class of division rings \(D\) that contains the division rings of quotients of the group algebras of torsion-free polycyclic-by-finite groups \(P\) over fields \(F\) of characteristic zero or sufficiently large in comparison with the Hirsch number \(h\) of \(P\). In this first paper of the series he concentrates on the soluble subgroups of \(GL(n,D)\). The Tits alternative is promised in Part II. With \(D\) as above let \(G\) be a soluble-by-finite subgroup of \(GL(n,D)\). The author gives a fairly precise description of \(G\). For example if \(G\) is completely reducible and \(D\) is the example above, then \(G\) is abelian, by finite, by soluble of derived length at most \(h\), by finite with order bounded in terms of \(n\) and \(h\). A very wide range of techniques is used, including some recent and yet to be published number-theoretic work of R. Murty stemming from his work on Artin's Problem. At one point (Theorem Iiii) the author needs to assume the generalised Riemann hypothesis. In my view the most novel and interesting innovation in the paper is the use of Krull dimension to bound the derived length of certain soluble groups in terms of the \(h\) above. The crucial result of the author here is the following. Let \(R\) be the ring of quotients of a crossed product of a Goldie \(F\)-algebra and a polycyclic-by-finite group with Hirsch number \(h\). Then the Krull dimension (say right) of the algebra \(R^{op}\otimes_ F R\) is at least \(h\). The author's proof uses R. Resco's theory of \(r\)-sequences, but direct proofs are now available.
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 16K40
Mathematics Subject Classification ID: 20G35
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 179079
zbMATH Keywords: subrings of matrix ring
zbMATH Keywords: division rings
zbMATH Keywords: division rings of quotients
zbMATH Keywords: group algebras of torsion-free polycyclic-by-finite groups
zbMATH Keywords: Hirsch number
zbMATH Keywords: soluble subgroups
zbMATH Keywords: Tits alternative
zbMATH Keywords: derived length
zbMATH Keywords: Krull dimension
zbMATH Keywords: crossed product
zbMATH Keywords: Goldie \(F\)-algebra
reviewed by: B. A. F. Wehrfritz
MaRDI profile type: Publication
full work available at URL: 0022-4049(93)90105-3
OpenAlex ID: W2072043843
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description: scientific article; zbMATH DE number 179079