Algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups

We study algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups. A special attention is paid to free nilpotent groups and the groups $\mathrm{UT}_n(\mathbb Z)$ of unitriangular$(n\times n)$-matrices over the ring $\mathbb Z$ of integers for arbitrary $n$. We observe that the sets of retracts of finitely generated nilpotent groups coincides with the sets of their algebraically closed subgroups. We give an example showing that a verbally closed subgroup in a finitely generated nilpotent group may fail to be a retract (in the case under consideration, equivalently, fail to be an algebraically closed subgroup). Another example shows that the intersection of retracts (algebraically closed subgroups) in a free nilpotent group may fail to be a retract (an algebraically closed subgroup) in this group. We establish necessary conditions fulfilled on retracts of arbitrary finitely generated nilpotent groups. We obtain sufficient conditions for the property of being a retract in a finitely generated nilpotent group. An algorithm is presented determining the property of being a retract for a subgroup in free nilpotent group of finite rank (a solution of a problem of Myasnikov). We also obtain a general result on existentially closed subgroups in finitely generated torsion-free nilpotent with cyclic center, which in particular implies that for each $n$ the group $\mathrm{UT}_n(\mathbb Z)$ has no proper existentially closed subgroups.