Divisible subgroups as exact equalizers in torsion-free nilpotent groups

Abstract

Let \(G\) be torsion-free nilpotent of class at most \(c\), and let \(H\leq G\) be divisible. We construct a torsion-free nilpotent group \(K\) of class at most \(c\) and embeddings \(f_0,f_1:G\hookrightarrow K\) whose equalizer is exactly \(H\). Consequently the dominion of \(H\) in \(G\), relative to torsion-free nilpotent groups of class at most \(c\), is \(H\), answering Kourovka Problem 17.34. The construction uses the rational Mal'cev correspondence and a derivation into a truncated quotient of the augmentation ideal of a universal enveloping algebra. The augmentation filtration preserves the original nilpotency-class bound. No finite-generation or normality hypothesis on \(H\) is required.