Generators of split extensions of Abelian groups by cyclic groups

TL;DR

This paper investigates the classification of generating tuples in split extensions of abelian groups by cyclic groups, providing a complete invariant for Nielsen equivalence and applying it to specific group classes.

Contribution

It introduces a new invariant for Nielsen equivalence in these groups and establishes a bijection with unimodular row orbits, extending understanding of generating sets in complex group structures.

Findings

Nielsen equivalence classes correspond to unimodular row orbits for infinite cyclic groups.

Complete invariant for Nielsen equivalence when the abelian subgroup is isomorphic to the ring R.

Classification results for Baumslag-Solitar, lamplighter, and split metacyclic groups.

Abstract

Let $G \simeq M \rtimes C$ 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 $GL_{n - 1}(R)$. 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.