The residual finiteness of ascending HNN-extensions of certain soluble groups

Abstract

If $G$ is a group with an injective endomorphism $\phi$, then the HNN-extension $G_\phi = \langle G, t \,:\, t^{-1}gt=g \phi \,\, \mbox{for all}\,\, g \in G\rangle $ is called the ascending HNN-extension of $G$ determined by $\phi$. We prove that $G_\phi$ is residually finite when $G$ is either finitely generated abelian-by-polycyclic-by-finite or reduced soluble-by-finite minimax. We also provide an example of a $3$-generator residually finite soluble group $G$ of derived length $3$ with a non-residually-finite ascending HNN-extension.