## 2007.69: On algebraic sets over metabelian groups

2007.69:
Vladimir Remeslennikov and Ralph Stöhr
(2005)
*On algebraic sets over metabelian groups.*
Journal of Group Theory, 8 (4).
pp. 491-513.
ISSN 1435-4446

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DOI: 10.1515/jgth.2005.8.4.491

## Abstract

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 |
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Subjects: | MSC 2000 > 17 Nonassociative rings and algebras MSC 2000 > 20 Group theory and generalizations |

MIMS number: | 2007.69 |

Deposited By: | Miss Louise Stait |

Deposited On: | 06 April 2007 |

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