Strongly verbally closed groups

TL;DR

This paper establishes conditions under which certain groups, including surface groups and free products satisfying a law, are algebraically closed in any group where they are verbally closed, extending previous results on free groups.

Contribution

It provides new sufficient conditions for groups to be algebraically closed in larger groups, especially for extensions of free groups by groups satisfying a law.

Findings

Fundamental groups of all closed surfaces (except Klein bottle) are algebraically closed.

Almost all free products of groups satisfying a law are algebraically closed.

Conditions are established for extensions of free groups to be algebraically closed.

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.