On algebraic sets over metabelian groups
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Vladimir Remeslennikov
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.
Walter de Gruyter GmbH & Co. KG
Articles in the same Issue
- Tri-extraspecial groups
- On commutators in p-groups
- Capability of nilpotent products of cyclic groups
- Products of characters and derived length. II
- Character degree graphs, blocks and normal subgroups
- Reflection triangles in Coxeter groups and biautomaticity
- On algebraic sets over metabelian groups
- Bounded automorphisms and quasi-isometries of finitely generated groups
- Pattern recognition and minimal words in free groups of rank 2
Articles in the same Issue
- Tri-extraspecial groups
- On commutators in p-groups
- Capability of nilpotent products of cyclic groups
- Products of characters and derived length. II
- Character degree graphs, blocks and normal subgroups
- Reflection triangles in Coxeter groups and biautomaticity
- On algebraic sets over metabelian groups
- Bounded automorphisms and quasi-isometries of finitely generated groups
- Pattern recognition and minimal words in free groups of rank 2
