The residual finiteness of ascending HNN-extensions of certain soluble groups
Let \(G\) be a group and \(\varphi\) an injective endomorphism of \(G\). The corresponding HNN-extension \(G_\varphi=\langle G,t\mid g^t=g^\varphi,\;g\in G\rangle\) is called an `ascending HNN-extension'. The authors consider the interesting question of when \(G_\varphi\) is residually finite. An obvious necessary condition is that \(G\) be residually finite, but this is not sufficient, even when \(G\) is a finitely generated soluble group of derived length \(3\), as the authors show. The main results of the paper assert that \(G_\varphi\) is residually finite if \(G\) is either finitely generated Abelian-by-polycyclic-by-finite or a reduced soluble-by-finite minimax group. It is left open whether \(G_\varphi\) is residually finite when \(G\) is a finitely generated residually finite, nilpotent-by-polycyclic-by-finite group.
scientific article; zbMATH DE number 1993648