Generators of split extensions of Abelian groups by cyclic groups

Abstract

Let $G\simeq M⋊C$ be an $n$-generator group with $M$ Abelian and $C$ cyclic. We study the Nielsen equivalence classes and T-systems of generating $n$-tuples of $G$. The subgroup $M$ can be turned into a finitely generated faithful module over a suitable quotient $R$ of the integral group ring of $C$. When $C$ is infinite, we show that the Nielsen equivalence classes of the generating $n$-tuples of $G$ correspond bijectively to the orbits of unimodular rows in $M^{n-1}$ under the action of a subgroup of $G{L}_{n-1}(R)$. Making no assumption on the cardinality of $C$, we exhibit a complete invariant of Nielsen equivalence in the case $M\simeq R$. As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag-Solitar groups, lamplighter groups and split metacyclic groups.

arXiv comment36 pages, The former Theorem F.ii has been retracted because the proof was wrong and couldn't be repaired. To appear in Groups, Geometry and Dynamics