Ascending HNN extensions of polycyclic groups have the same cohomology as their profinite completions

Abstract

Assume $G$ is a polycyclic group and $\phi:G\to G$ an endomorphism. Let $G\ast_{\phi}$ be the ascending HNN extension of $G$ with respect to $\phi$; that is, $G\ast_{\phi}$ is given by the presentation $$G\ast_{\phi}= < G, t \ |\ t^{-1}gt = \phi(g)\ \{for all}\ g\in G >.$$ Furthermore, let $\hat{G\ast_{\phi}}$ be the profinite completion of $G\ast_{\phi}$. We prove that, for any finite discrete $\hat{G\ast_{\phi}}$-module $A$, the map $H^*(\hat{G\ast_{\phi}}, A)\to H^*(G\ast_{\phi},A)$ induced by the canonical map $G\ast_{\phi}\to \hat{G\ast_{\phi}}$ is an isomorphism.