On conditions for the approximability of the fundamental groups of graphs of groups by root classes of groups

TL;DR

This paper investigates conditions under which the fundamental groups of graphs of groups can be approximated by root classes of groups, extending known residual properties from vertex groups to edge subgroups.

Contribution

It generalizes residual $ extit{C}$-group conditions from vertex groups to edge subgroups in graphs of groups, and discusses limitations for periodic groups.

Findings

Replacing vertex groups with edge subgroups preserves residual properties.

The converse implication fails for certain periodic group classes.

Provides conditions for approximability of fundamental groups by root classes.

Abstract

Suppose that $\Gamma$ is a non-empty connected graph, $\mathfrak{G}$ is the fundamental group of a graph of groups over $\Gamma$, and $\mathcal{C}$ is a root class of groups (the last means that $\mathcal{C}$ contains non-trivial groups and is closed under taking subgroups, extensions, and Cartesian powers of a certain type). It is known that $\mathfrak{G}$ is residually a $\mathcal{C}$-group if it has a homomorphism onto a group from $\mathcal{C}$ acting injectively on all vertex groups. We prove that, in this assertion, the words "vertex groups" can be replaced by "edge subgroups" provided all vertex groups are residually $\mathcal{C}$-groups. We also show that the converse doesn't need to hold if $\mathcal{C}$ consists of periodic groups and contains at least one infinite group.