Varietal product

Definition
Suppose $$\mathcal{V}$$ is a subvariety of the variety of groups that contains the subvariety of abelian groups. Suppose $$G_i, i \in I$$ are groups (not necessarily in $$\mathcal{V}$$). The varietal product of these groups with respect to the variety $$\mathcal{V}$$, also called the verbal product with respect to $$\mathcal{V}$$, is defined as follows:


 * Suppose $$G$$ is the external free product of the groups $$G_i$$ with inclusion maps $$\varphi_i:G_i \to G$$.
 * Let $$V(G)$$ denote the verbal subgroup of $$G$$ corresponding to the variety $$\mathcal{V}$$.
 * Let $$W$$ be the normal closure in $$G$$ of the join of all the commutator subgroups $$[\varphi_i(G_i),\varphi_j(G_j)]$$.
 * The varietal product is defined as $$G/(V(G) \cap W)$$.