Strongly verbally closed groups

Abstract

It was recently proven that all free and many virtually free verbally closed subgroups are algebraically closed in any group. We establish sufficient conditions for a group that is an extension of a free non-abelian group by a group satisfying a non-trivial law to be algebraically closed in any group in which it is verbally closed. We apply these conditions to prove that the fundamental groups of all closed surfaces (except the Klein bottle) and almost all free products of groups satisfying a non-trivial law are algebraically closed in any group in which they are verbally closed.