Geodesic
Properties and universal theories for partially commutative nilpotent metabelian groups
Algebra i logika, Tome 51 (2012) no. 4, pp. 429-457.

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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.
Keywords: partially commutative nilpotent metabelian groups, annihilator, centralizer, graph of group, tree, universal theory. 
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Ch. K. Gupta; E. I. Timoshenko. Properties and universal theories for partially commutative nilpotent metabelian groups. Algebra i logika, Tome 51 (2012) no. 4, pp. 429-457. http://geodesic.mathdoc.fr/item/AL_2012_51_4_a1/

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