Geodesic
Existentially closed subgroups of free nilpotent groups
Algebra i logika, Tome 53 (2014) no. 1, pp. 45-59.

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Let $\mathcal N_c$ be a variety of all nilpotent groups of class at most $c$, and let $N_{r,c}$ be a free nilpotent group of finite rank $r$ and nilpotency class $c$. It is proved that a subgroup $N$ of $N_{r,c}$ for $c\ge3$ is existentially closed in $N_{r,c}$ iff $N$ is a free factor of the group $N_{r,c}$ with respect to the variety $\mathcal N_c$. Consequently, $N\simeq N_{m,c}$, $1\le m\le r$ and $m\ge c-1$.
Keywords: existentially closed subgroup, free nilpotent group, discriminating extension. 
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V. A. Roman'kov; N. G. Khisamiev. Existentially closed subgroups of free nilpotent groups. Algebra i logika, Tome 53 (2014) no. 1, pp. 45-59. http://geodesic.mathdoc.fr/item/AL_2014_53_1_a3/

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