Groups with the same cohomology as their profinite completions

TL;DR

This paper investigates groups whose low-dimensional cohomology matches that of their profinite completions, describing specific constructions within these classes and examining residual finiteness properties.

Contribution

It characterizes certain free products with amalgam and HNN extensions within classes of groups sharing cohomology with their profinite completions and studies residual finiteness in these classes.

Findings

Certain free products with amalgam are in classes _n.

Some HNN extensions belong to these classes.

Residual finiteness is analyzed for groups in _2.

Abstract

For any positive integer $n$, $\mathcal{A}_n$ is the class of all groups $G$ such that, for $0\leq i\leq n$, $H^i(\hat{G},A)\cong H^i(G,A)$ for every finite discrete $\hat{G}$-module $A$. We describe certain types of free products with amalgam and HNN extensions that are in some of the classes $\mathcal{A}_n$. In addition, we investigate the residually finite groups in the class $\mathcal{A}_2$.