On the subgroup separability of the free product of groups

TL;DR

This paper establishes a criterion for subgroup separability in free products of residually  groups, linking subgroup properties to group varieties and providing a method to describe -separable subgroups.

Contribution

It introduces a new criterion for -separability of subgroups in free products, extending understanding of subgroup structure in complex group constructions.

Findings

Provides a criterion for -separability of subgroups satisfying a non-trivial identity.

Describes -separable -subgroups in free products based on known subgroup descriptions.

Connects subgroup separability to group varieties and their unions.

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 $A_{i}$ ($i \in \mathcal{I}$).