Verbally and existentially closed subgroups of free nilpotent groups

Abstract

We study verbally closed subgroups of free solvable groups. A number of results is proved that give sufficient conditions under whose a verbally closed subgroup is turned to be a retract and so algebraically closed of the full group.

On free decompositions of verbally closed subgroups in free products of finite groups

“…It is easy to prove (see, for instance, [11]) that, if H is a retract of G, then H is algebraically (and verbally) closed in G. If G is a finitely presented group, and H is a finitely generated subgroup of G, then H is an algebraically closed subgroup of G if and only if H is a retract of G (see [11]). It was proved in [11] and [13] that a subgroup H of a free or free nilpotent group G of finite rank is verbally closed if and only if H is a retract of G. This implies that the properties of being verbally closed, being algebraically closed, and being a retract are equivalent for subgroups of free or free nilpotent groups of finite rank. Also in [11] the following result was obtained.…”

Finite and nilpotent strongly verbally closed groups

“…Algebraic closedness is a stronger property than verbal closedness; however these properties turn out to be equivalent in many cases (see [Rom12], [RKh13], [MR14], [Mazh17], [RKhK17], [KM18], [KMM18], [Mazh18], [Bog18], [Bog19], [Mazh19], [RT19], [RT20], [Tim21]). A group H is called strongly verbally closed [Mazh18] if it is algebraically closed in any group containing H as a verbally closed subgroup.…”

“…Proof. If a group H is algebraically closed in a variety M (i. e. algebraically closed in any group from M that contains H), then H is strongly verbally closed by the lemma from [RKh13] mentioned above. Thus, the embedding theorem follows from Scott's theorem [Sco51]:…”

The Klein bottle group is not strongly verbally closed, though awfully close to being so

“…A subgroup H of a group G is called verbally closed [MR14] (see also [Rom12], [RKh13], [Mazh17], [KlMa18], [KMM18], [Mazh18], [Bog18], [Bog19], [Mazh19], [RT19]) if any equation of the form w(x, y, . .…”