In the present work, we investigate the Lie algebra of the Formanek-Procesi group $\textrm {FP}(A_{\Gamma })$ with base group $A_{\Gamma }$ a right-angled Artin group. We show that the Lie algebra $\textrm {gr}(\textrm {FP}(A_{\Gamma }))$ has a presentation that is dictated by the group presentation. Moreover, we show that if the base group $G$ is a finitely generated residually finite $p$-group, then $\textrm { FP}(G)$ is residually nilpotent. We also show that $\textrm {FP}(A_{\Gamma })$ is a residually torsion-free nilpotent group.

Let G be a Baumslag–Solitar group. We calculate the intersection $\gamma_{\omega}(G)$ of all terms of the lower central series of G. Using this, we show that $[\gamma_{\omega}(G),G]=\gamma_{\omega}(G)$, thus answering a question of Bardakov and Neschadim [1]. For any $c \in \mathbb{N}$, with $c \geq 2$, we show, by using Lie algebra methods, that the quotient group $\gamma_{c}(G)/\gamma_{c+1}(G)$ of the lower central series of G is finite.

We give an algorithm which for a given set of generators can decide whether a stem product of infinite cyclic groups is a one-relator group. We also generalize this to the case of one-relator products of groups.

We investigate asphericity of the relative group presentation $$\langle G,t\mid atbtctdtet=1\rangle$$ and prove it aspherical provided thesubgroup of $G$ generated by $$\{ ab^{-1}, bc^{-1}, cd^{-1}, de^{-1}\}$$ is neither finite cyclic nor a finite triangle group. We also prove a similar result for the closely related relative group presentation$$\langle G,s,t \mid \alpha s\beta s\gamma t=1=\delta t\varepsilon t\zetas^{-1}\rangle. 2000 Mathematics Subject Classification: 20F05, 57M05.