On algebraic sets over metabelian groups

Remeslennikov, Vladimir and Stöhr, Ralph (2005) On algebraic sets over metabelian groups. Journal of Group Theory, 8 (4). pp. 491-513. ISSN 1435-4446

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We investigate algebraic sets over certain finitely generated torsion-free metabelian groups. The class of groups under consideration is the class of so-called ρ-groups. It consists of all wreath products of finitely generated free abelian groups and their subgroups. In particular, it includes all free metabelian groups of finite rank. Our main result is a characterization of certain irreducible algebraic sets over ρ-groups. More precisely, we consider irreducible algebraic sets which are determined by a system of equations in n indeterminates. For their coordinate groups, we introduce a discrete invariant called the relative characteristic. This is an ordered pair of non-negative integers. We determine the structure of the coordinate group of the n-dimensional affine space, and show that its relative characteristic is (n, n). Then we characterize the irreducible algebraic sets of relative characteristic (n, n) and (0, k ) where 0 ≤ k ≤ n . We also obtain some examples of somewhat unusual algebraic sets over ρ-groups, thus demonstrating that algebraic sets over these groups are much more varied and complicated than, say, algebraic sets over free groups.

Item Type: Article
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 17 Nonassociative rings and algebras
MSC 2010, the AMS's Mathematics Subject Classification > 20 Group theory and generalizations
Depositing User: Ms Lucy van Russelt
Date Deposited: 06 Apr 2007
Last Modified: 20 Oct 2017 14:12
URI: https://eprints.maths.manchester.ac.uk/id/eprint/771

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