Residual properties of groups defined by basic commutators

TL;DR

This paper investigates the residual properties of groups generated by basic commutators, establishing conditions under which they are residually torsion-free nilpotent and providing counterexamples.

Contribution

It proves that Hydra groups and certain generalizations are residually torsion-free nilpotent, and presents an example of a group that is not, expanding understanding of these properties.

Findings

Hydra groups are residually torsion-free nilpotent

Certain quotients of groups by basic commutators share residual properties

An example of a group not residually torsion-free nilpotent is provided

Abstract

In this paper we study the residual nilpotence of groups defined by basic commutators. We prove that the so-called Hydra groups as well as certain of their generalizations and quotients are, in the main, residually torsion-free nilpotent. By way of contrast we give an example of a group defined by two basic commutators which is not residually torsion-free nilpotent.