On the Residual Finiteness of Descending HNN-Extensions of Groups

Let $G$ be a group of finite generic rank, $\varphi $ an injective endomorphism of the group $G$, and $G(\varphi)$ the descending HNN-extension of $G$ corresponding to the endomorphism $\varphi$. Let the index of the subgroup $G\varphi$ in $G$ be finite and equal to $n$. It is proved that, if the group $G$ is almost residually $\pi$-finite for some set $\pi$ of primes coprime to $n$, then the group $G(\varphi)$ is residually finite. This generalizes a series of known results, including the Wise–Hsu theorem on the residual finiteness of an arbitrary descending HNN-extension of any almost polycyclic group.