HNN-extensions of finitely presented soluble groups
The author studies some properties of finitely presented soluble groups. He first considers HNN-extensions of such groups, and proves that if \(G\) is any finitely presented nilpotent-by-abelian-by-finite group and \(H\) is any descending HNN-extension of \(G\), then \(H\) is also nilpotent-by- abelian-by-finite. The proof of this result consists largely of commutative algebra. With similar methods he proves that every soluble group of automorphisms of a finitely presented nilpotent-by-abelian-by- finite group is likewise nilpotent-by-abelian-by-finite. This result should be related to the well-known theorem of Mal'cev-Baer that every soluble group of automorphisms of a polycyclic-by-finite group is polycyclic. The author also points out that the above theorems remain true if the hypothesis that the group is finitely presented is replaced by the assumption that it has finite Prüfer rank.
scientific article; zbMATH DE number 554193