Free polynilpotent groups and the Magnus property

TL;DR

This paper characterizes when free polynilpotent and related soluble groups have the Magnus property, showing it holds only for certain nilpotency classes, and constructs specific examples with this property.

Contribution

It provides a complete characterization of the Magnus property in free polynilpotent groups and introduces new techniques for analyzing this property in soluble groups.

Findings

Free polynilpotent groups have the Magnus property iff they are nilpotent of class at most 2.

Constructed examples of nilpotent groups with the Magnus property for various classes.

Developed methods to establish or disprove the Magnus property in soluble groups.

Abstract

Motivated by a classic result for free groups, one says that a group $G$ has the Magnus property if the following holds: whenever two elements generate the same normal subgroup of $G$, they are conjugate or inverse-conjugate in $G$. It is a natural problem to find out which relatively free groups display the Magnus property. We prove that a free polynilpotent group of any given class row has the Magnus property if and only if it is nilpotent of class at most $2$. For this purpose we explore the Magnus property more generally in soluble groups, and we produce new techniques, both for establishing and for disproving the property. We also prove that a free centre-by-(polynilpotent of given class row) group has the Magnus property if and only if it is nilpotent of class at most $2$. On the way, we display $2$-generated nilpotent groups (with non-trivial torsion) of any prescribed nilpotency class with the Magnus property. Similar examples of finitely generated, torsion-free nilpotent groups are hard to come by, but we construct a $4$-generated, torsion-free, class-$3$ nilpotent group of Hirsch length $9$ with the Magnus property. Furthermore, using a weak variant of the Magnus property and an ultraproduct construction, we establish the existence of metabelian, torsion-free, nilpotent groups of any prescribed nilpotency class with the Magnus property.