On the Cantor-Bendixson rank of metabelian groups. (English. French summary) Zbl 1238.20049
If \(G\) is a group, its (intrinsic) Cantor-Bendixson rank is the Cantor-Bendixson rank of the trivial subgroup in the space of normal subgroups of \(G\), equipped with the Chabauty topology. The author proves that for any ordinal \(\alpha<\omega^\omega\), there exists a \(2\)-generated finitely presented metabelian-by-(finite cyclic) group of Cantor-Bendixson rank \(\alpha\). He further gives exact computations of the Cantor-Bendixson rank of finitely generated metabelian groups, and also considers the Cantor-Bendixson rank of wreath products of finitely generated groups.
Reviewer: Eric Jaligot (Grenoble)
MSC:
| 20F05 | Generators, relations, and presentations of groups |
| 20F16 | Solvable groups, supersolvable groups |
| 20E15 | Chains and lattices of subgroups, subnormal subgroups |
| 13E05 | Commutative Noetherian rings and modules |
| 57M07 | Topological methods in group theory |
| 03C45 | Classification theory, stability, and related concepts in model theory |
Keywords:
finitely generated metabelian groups; space of marked groups; Cantor-Bendixson analysis; Bieri-Strebel invariants; lattices of subgroups; finitely presented groupsReferences:
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