Certain residual properties of generalized Baumslag–Solitar groups

Abstract

Let G be a generalized Baumslag–Solitar group, and let C be a class of groups containing at least one non-trivial group and closed under taking subgroups, extensions, and Cartesian products of the form ∏y∈YXy, where X, Y∈C and Xy is an isomorphic copy of X for every y∈Y. We give a criterion for G to be residually a C-group provided C consists only of periodic groups. We also prove that G is residually a torsion-free C-group if C contains at least one non-periodic group and is closed under taking homomorphic images. These facts generalize and strengthen some known results. We also provide criteria for a GBS-group to be a) residually nilpotent; b) residually torsion-free nilpotent; c) residually free.