Cohomological Separability of Baumslag--Solitar groups and Their Generalisations

TL;DR

This paper investigates when Baumslag--Solitar groups and their generalizations have cohomology that is preserved under profinite completion, providing explicit criteria and revealing new examples with unique properties.

Contribution

It characterizes cohomological separability for generalized Baumslag--Solitar groups and classifies Baumslag--Solitar groups into three types based on this property and cohomological dimension.

Findings

Provides explicit conditions for cohomological separability in generalized Baumslag--Solitar groups.

Classifies Baumslag--Solitar groups into three categories based on cohomological properties.

Identifies examples of non-residually-finite groups with separable cohomology.

Abstract

A group $\Gamma$ has separable cohomology if the profinite completion map $\iota \colon \Gamma \to \widehat{\Gamma}$ induces an isomorphism on cohomology with finite coefficient modules. In this article, cohomological separability is decided within the class of generalised Baumslag--Solitar groups, i.e. graphs of groups with infinite cyclic fibers. Equivalent conditions are given both explicitly in terms of the defining graph of groups and in terms of the induced topology on vertex groups. Restricted to the class of Baumslag--Solitar groups, we obtain a trichotomy of cohomological separability and cohomological dimension of the profinite completions. In particular, this yields examples of non-residually-finite one-relator groups which have separable cohomology, and examples which do not.