Free products of groups are strongly verbally closed

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.