Generators of split extensions of Abelian groups by cyclic groups

2016/04/29 by Luc Guyot, Guyot, Luc
Mathematics · #20F05 (Primary) #20F16 #20F28 (Secondary) #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20F05 #msc:20F16 #msc:20F28

paper · pdf · doi:10.48550/arxiv.1604.08896

36 pages, The former Theorem F.ii has been retracted because the proof was wrong and couldn't be repaired.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

arxiv created 2018/06/22 · arxiv updated 2018/06/25

Abstract

Let G ≃ 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 ≃ R. As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag-Solitar groups, lamplighter groups and split metacyclic groups.

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