Free products of groups are strongly verbally closed

TL;DR

This paper proves that any non-trivial free product of groups is algebraically closed in any group where it is verbally closed, extending previous results on virtually free and surface groups.

Contribution

It establishes that all non-trivial free products are strongly verbally closed, a significant generalization of prior work on specific classes of groups.

Findings

Non-trivial free products are algebraically closed in any group where they are verbally closed.

Extends known results from virtually free and surface groups to all non-trivial free products.

Provides a unifying framework for understanding algebraic closure properties of free product groups.

Abstract

In a number of recent works, it has been established that many virtually free groups, almost all fundamental groups of surfaces and all groups which are nontrivial free products of groups satisfying a non-trivial law are algebraically closed in any group in which they are verbally closed. In this work we establish that any group which is a non-trivial free product is algebraically closed in any group in which it is verbally closed.