Abstract:

The article proves the equivalence of the properties of being equationally Noethericity of a partially commutative two-step nilpotent group $G$ and being equationally Noethericity of its commutative graph $\Gamma_G$, considered in the category of graphs with loops. In particular, it is shown that to study the equationally Noethericity of a partially commutative two-step nilpotent group, it is sufficient to consider equations in one variable over this group. Using the concept of residualizability of groups, we prove that an arbitrary partially commutative two-step nilpotent group can be embedded in a countable direct power of a free two-step nilpotent group of rank $2$. In addition, earlier in the works of A.J. Duncan, I.V. Kazachkov and V.N. Remeslennikov it was shown that the centralizer dimension of each finitely generated free partially com\-mutative group coincides with the height of the lattice of canonical centralizers of this group. In our work we present some results relating the centralizer dimension and the height of the lattice of canonical centralizers for the case of an infinitely generated free partially commutative group and generalize a similar result from the work of V. Blatherwick to the case of an infinitely generated partially commutative two-step nilpotent group.