On residually nilpotence of group extensions

TL;DR

This paper investigates conditions under which extensions of residually nilpotent groups remain residually nilpotent, with specific focus on semi-direct products involving free groups and infinite cyclic groups, providing new criteria and structural insights.

Contribution

It establishes sufficient conditions for residual nilpotence in group extensions and characterizes the lower central series length for certain semi-direct products.

Findings

Identifies conditions for residual nilpotence in group extensions.

Determines the length of the lower central series for specific semi-direct products.

Shows for $n=2$, the series length is 2, $\omega$, or $\omega^2$.

Abstract

We study the following question: under what conditions extension of one residually nilpotent group by another residually nilpotent group is residually nilpotent? We prove some sufficient conditions under which this extension is residually nilpotent. Also, we study this question for semi-direct products and, in particular, for extensions of free group by infinite cyclic group: $F_n \rtimes_{\varphi} \mathbb{Z}$. We find conditions under which this group is residually nilpotent, find conditions under which this group has long lower central series. In particular, we prove that for $n=2$ the length of the lower central series of $F_n \rtimes_{\varphi} \mathbb{Z}$ is equal to 2, $\omega$ or $\omega^2$.