Properties and universal theories for partially commutative nilpotent metabelian groups

Partially commutative nilpotent metabelian groups are considered. We describe how annihilators of elements of the commutator subgroup of a group $G$, as well as centralizers of elements of $G$ in its commutator subgroup $G'$, are structured. It turns out that in the case where a defining graph of a group is a tree, the intersection of centralizers of distinct vertices and $G'$ coincides with the last nontrivial commutator subgroup of $G$. Universal theories for partially commutative nilpotent metabelian groups are compared: conditions on defining graphs of two partially commutative nilpotent metabelian groups are formulated which are sufficient for the two groups to have equal universal theories; conditions on defining graphs of two partially commutative metabelian groups are specified which are sufficient for the two groups to be universally equivalent; a criterion is given that decides whether two partially commutative nilpotent metabelian groups defined by trees are universally equivalent.