Geodesic
Generators of split extensions of Abelian groups by cyclic groups
Luc Guyot1
1Ecole Polythechnique Fédérale de Lausanne, Genève, Switzerland
Groups, geometry, and dynamics, Tome 12 (2018) no. 2, pp. 765-802

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DOI

Let G≃M⋊C be an n-generator group which is a split extension of an Abelian group M by a cyclic group C. 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 Mn−1 under the action of a subgroup of GLn−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, split metacyclic groups and lamplighter groups.
DOI : 10.4171/ggd/455
Classification : 20-XX
Mots-clés : Nielsen equivalence, T-systems, Abelian-by-cyclic groups, lamplighter groups, Baumslag–Solitar groups, metacyclic groups, Laurent polynomials, generalized Euclidean rings, quasi-Euclidean rings, special Whitehead group

Luc Guyot  1

1 Ecole Polythechnique Fédérale de Lausanne, Genève, Switzerland 
Luc Guyot. Generators of split extensions of Abelian groups by cyclic groups. Groups, geometry, and dynamics, Tome 12 (2018) no. 2, pp. 765-802. doi: 10.4171/ggd/455
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     year = {2018},
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     number = {2},
     doi = {10.4171/ggd/455},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/455/}
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