Separability Properties of Nilpotent $\mathbb{Q}[x]$-Powered Groups

Abstract

In this paper we study conjugacy and subgroup separability properties in the class of nilpotent $\mathbb{Q}[x]$-powered groups. Many of the techniques used to study these properties in the context of ordinary nilpotent groups carry over naturally to this more general class. Among other results, we offer a generalization of a theorem due to G. Baumslag. The generalized version states that if $G$ is a finitely $\mathbb{Q}[x]$-generated $\mathbb{Q}[x]$-torsion-free nilpotent $\mathbb{Q}[x]$-powered group and $H$ is a $\mathbb{Q}[x]$-isolated subgroup of $G,$ then for any prime $\pi \in \mathbb{Q}[x]$, ${\bigcap }_{i=1}^{\infty }{G}^{{\pi }^i}H=H.$

14 pages; Submitted to the proceedings of the Elementary Theory of Groups Conference 2018 honoring Ben Fine and Anthony Gaglione on the occasion of their birthdays; and to Gilbert Baumslag (in memoriam)