Baumslag-Solitar groups and residual nilpotence

TL;DR

This paper investigates the structure of Baumslag-Solitar groups by analyzing their lower central series, revealing their residual properties and finite quotients, and answering a previously posed question.

Contribution

It provides explicit calculations of the lower central series intersections and shows their properties, including residual nilpotence and finiteness of certain quotients.

Findings

The intersection of all lower central series terms equals its own commutator with the group.

The quotient groups of consecutive lower central series are finite.

The paper answers a question posed by Bardakov and Neschadim.

Abstract

For a Baumslag-Solitar group $G$ we calculate the intersection $\gamma_w(G)$ of all terms of the lower central sequence of $G$.Using this we are able to show that $[\gamma_w(G),G]=\gamma_w(G)$ thus answering a question of Bardakov and Neschadim. Finally we show that the quotient groups $\gamma_c(G)/\gamma_{c+1}(G)$ of the lower central series of $G$ are finite.