JournalsggdVol. 12, No. 2pp. 765–802

Generators of split extensions of Abelian groups by cyclic groups

  • Luc Guyot

    Ecole Polythechnique Fédérale de Lausanne, Genève, Switzerland
Generators of split extensions of Abelian groups by cyclic groups cover
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Abstract

Let GMCG \simeq M \rtimes C be an nn-generator group which is a split extension of an Abelian group MM by a cyclic group CC. We study the Nielsen equivalence classes and T-systems of generating nn-tuples of GG. The subgroup MM can be turned into a finitely generated faithful module over a suitable quotient RR of the integral group ring of CC. When CC is infinite, we show that the Nielsen equivalence classes of the generating nn-tuples of GG correspond bijectively to the orbits of unimodular rows in Mn1M^{n -1} under the action of a subgroup of GLn1(R)_{n - 1}(R). Making no assumption on the cardinality of CC, we exhibit a complete invariant of Nielsen equivalence in the case MRM \simeq R. As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag–Solitar groups, split metacyclic groups and lamplighter groups.

Cite this article

Luc Guyot, Generators of split extensions of Abelian groups by cyclic groups. Groups Geom. Dyn. 12 (2018), no. 2, pp. 765–802

DOI 10.4171/GGD/455