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

2023/03/17 by Sokolov, E. V. · 1 citation
#20E08 (Primary) 20E06 (Secondary) #20E26 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2303.09815

Abstract

Suppose that Γ is a non-empty connected graph, \mathfrakG is the fundamental group of a graph of groups over Γ, and C is a root class of groups (the last means that C contains non-trivial groups and is closed under taking subgroups, extensions, and Cartesian powers of a certain type). It is known that \mathfrakG is residually a C-group if it has a homomorphism onto a group from 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 C-groups. We also show that the converse doesn't need to hold if C consists of periodic groups and contains at least one infinite group.

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