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

TL;DR

This paper proves that for ascending HNN extensions of polycyclic groups, the cohomology with finite coefficients is the same as that of their profinite completions, highlighting a deep connection between algebraic and topological properties.

Contribution

It establishes an isomorphism between the cohomology of ascending HNN extensions of polycyclic groups and their profinite completions for finite modules, a novel result in group cohomology.

Findings

Cohomology groups of ascending HNN extensions match those of their profinite completions.

The canonical map induces an isomorphism in cohomology for finite modules.

Supports the understanding of the relationship between algebraic and profinite structures.

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.