$G$-identities and $G$-varieties

G. Baumslag, A. Myasnikov and Remeslennikov [J. Algebra 219 (1999), no. 1, 16–79; MR1707663 (2000j:14003)] presented the fundamentals of algebraic geometry over a fixed group $G$; in particular, they introduced the concept of a category of $G$-groups. For groups in this category, one can also define the concepts of $G$-identity and $G$-variety. We present the fundamentals of the theory of varieties in the category of $G$-groups, of which the most essential is the concept of the group $V_{n,\mathrm{red}}(G)$ of reduced $G$-identities of rank $n$, which is important for the computation of the coordinate groups for algebraic sets over $G$. We prove that $V_{n,\mathrm{red}}(G)=1$ for all natural numbers $n$ if $G$ is a group that is close to a free or relatively free group for some variety of nilpotent groups of rank not less than the nilpotency class of $G$.