The Root Class Residuality of the Tree Product of Groups with Amalgamated Retracts

Abstract

Given a root class \(\mathscr{K}\) of groups, we prove that the tree product of residually \(\mathscr{K}\)-groups with amalgamated retracts is a residually \(\mathscr{K}\)-group. This yields a criterion for the \(\mathscr{K}\)-residuality of Artin and Coxeter groups with tree structure. We also prove that the HNN-extension X of a residually \(\mathscr{K}\)-group B is a residually \(\mathscr{K}\)-group provided that the associated subgroups of X are retracts in B and \(\mathscr{K}\) contains at least one nonperiodic group.