This paper aims to introduce and explore the concept of Lie perfect multiplicative Lie algebras, with a particular focus on their connections to the central extension theory of multiplicative Lie algebras. The primary objective is to establish and provide proof for a range of results derived from Lie perfect multiplicative Lie algebras. Furthermore, the study extends the notion of Lie nilpotency by introducing and examining the concept of local nilpotency within multiplicative Lie algebras. The...

We describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon’s completion $G^{\mathbb{Z}\left[t\right]}$ of the group $G$, or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of $G^{\mathbb{Z}\left[t\right]}$ containing $G$ is universally equivalent to $G$. Since finitely generated...

When localizing the semidirect product of two groups, the effect on the factors is made explicit. As an application in Topology, we show that the loop space of a based connected CW-complex is a P-local group, up to homotopy, if and only if π1X and the free homotopy groups [Sk-1, ΩX], k ≥ 2, are P-local.