On conditions for the root-class residuality of the fundamental groups of graphs of groups

E. V. Sokolov

Published 2023-03-17Version 1

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 of $\mathcal{C}$ acting injectively on all the vertex groups. We prove that, in this assertion, the words "vertex groups" can be replaced by "edge subgroups". 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.