Arithmeticity of stabilizers of ideals in group rings
type: Item
instance of: scholarly article
title: Arithmeticity of stabilizers of ideals in group rings
zbMATH Open document ID: 0532.16006
DOI: 10.1007/BF01403091
published in: Inventiones Mathematicae
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143089
review text: A conjecture of Bachmuth, Baumslag, Dyer and Mochizuki states that if G is a two-generator metabelian group then the image of Aut(G) in \(Aut(G_{ab})\) is finitely presented. This can be reduced to the question of whether the stabilizer in GL(2,\({\mathbb{Z}})\) of an ideal in \({\mathbb{Z}}[x,x^{-1},y,y^{-1}]\) is finitely presented. This question is answered in the affirmative. This is done by first proving the main theorem: if A is the free Abelian group of finite rank n\(>1\) and I an ideal of either of the group rings \({\mathbb{Q}}[A]\) or \({\mathbb{Z}}[A]\) such that the stabilizer Stab(I) in GL(n,\({\mathbb{Z}})\) and all its subgroups of finite index are irreducible subgroups of GL(n,\({\mathbb{Q}})\) (i.e. Stab(I) and all its subgroups of finite index act rationally irreducibly on A) then Stab(I) is commensurable with an arithmetic subgroup of GL(n,\({\mathbb{Z}})\). (Two subgroups are commensurable if their intersection has finite index in each.) Given this theorem it is easy to deduce that Stab(I) is finitely presented if I is an ideal of \({\mathbb{Z}}[x,x^{-1},y,y^{-1}]\). For, if Stab(I) satisfies the hypothesis of the main theorem then it is arithmetic and hence finitely presented by a result of Borel and Harish- Chandra [see \textit{A. Borel}, Proc. Internat. Congr. Math. Stockholm 1962, 10-22 (1963; Zbl 0134.165)] and if not then Stab(I) has a subgroup H of finite index which leaves a nontrivial subspace of \({\mathbb{Q}}^ 2\) invariant, H is solvable and hence polycyclic by a theorem of \textit{A. I. Mal'tsev} [Mat. Sb., Nov. Ser. 28(70), 567-588 (1951; Zbl 0043.023); transl. in Am. Math. Soc. Transl., II. Ser. 2, 1-21 (1956)] so again Stab(I) is finitely presented. The starting point for the proof of the main theorem is a result of \textit{G. M. Bergman} [Trans. Am. Math. Soc. 157, 459-470 (1971; Zbl 0197.17102)] which states that if I is an ideal of \({\mathbb{Z}}[A]\) satisfying the hypotheses of the main theorem then either \(I=0\) or \(\dim_…
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20K99
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 20H05
zbMATH DE Number: 3843939
zbMATH Keywords: ideal of group rings
zbMATH Keywords: two-generator metabelian group
zbMATH Keywords: stabilizer
zbMATH Keywords: irreducible subgroups
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: arithmetic subgroup
zbMATH Keywords: arithmetic groups
zbMATH Keywords: prime ideals
zbMATH Keywords: semiprime ideal
author: Daniel R. Farkas
author: Robert L. Snider
reviewed by: Patrick F. Smith
MaRDI profile type: Publication
cites work: Automorphism groups of finitely generated nilpotent groups
cites work: Automorphism Groups of Two Generator Metabelian Groups
cites work: Q5651360
cites work: The Logarithmic Limit-Set of an Algebraic Variety
cites work: Q5612538
cites work: Primitivity in representations of polycyclic groups
cites work: Prime Ideals in Group Rings of Polycyclic Groups
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description: scientific article; zbMATH DE number 3843939