On the subgroup separability of the free product of groups
E. V. Sokolov
TL;DR
This work studies $\\mathcal{C}$-separability of subgroups in a free product $G=\\ast_{i\\in\\mathcal{I}} A_i$ with $\\mathcal{C}$ a root class and each $A_i$ residually $\\mathcal{C}$. A central theorem gives a crisp equivalence: a subgroup $H\\le G$ that satisfies a nontrivial identity is $\\mathcal{C}$-defective in $G$ if and only if it is conjugate to a $\\mathcal{C}$-defective subgroup of some $A_i$, which yields a practical criterion for $\\mathcal{C}$-separability. Consequences describe $\\mathcal{C}$-defective and $\\mathcal{V}$-subgroups of $G$ via the corresponding subgroups of the factors, and show that if each $A_i$ has the relevant separability property for all $\\mathfrak{P}(\\mathcal{C})^{\\prime}$-isolated $\\mathcal{V}$-subgroups (or cyclic subgroups), then $G$ inherits it. The results generalize classical theorems on free products and provide a framework for transferring explicit separability descriptions from factors to the whole free product.
Abstract
Suppose that $\mathcal{C}$ is a root class of groups (i.e., a class of groups that contains non-trivial groups and is closed under taking subgroups and unrestricted wreath products), $G$ is the free product of residually $\mathcal{C}$-groups $A_{i}$ ($i \in \mathcal{I}$), and $H$ is a subgroup of $G$ satisfying a non-trivial identity. We prove a criterion for the $\mathcal{C}$-separability of $H$ in $G$. It follows from this criterion that, if $\{\mathcal{V}_{j} \mid j \in \mathcal{J}\}$ is a family of group varieties, each $\mathcal{V}_{j}$ ($j \in \mathcal{J}$) is distinct from the variety of all groups, and $\mathcal{V} = \bigcup_{j \in \mathcal{J}} \mathcal{V}_{j}$, then one can give a description of $\mathcal{C}$-separable $\mathcal{V}$-subgroups of $G$ provided such a description is known for every group $A_{i}$ ($i \in \mathcal{I}$).