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

Abstract

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

arXiv commentThe second, third, and fourth versions improve on the exposition in the first version. The fifth version includes a proof that every residually finite, solvable, minimax group is "good."