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

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

2010-09-14

arxiv.org/abs/1009.2645v5

Mathematics

Group Theory

The second, third, and fourth versions improve on the exposition in the first version. The fifth version includes a proof that

Scientific paper

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.

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