Generalized torsion, unique root property and Baumslag--Solitar relation for knot groups

TL;DR

This paper investigates the properties of knot groups related to generalized torsion and the Baumslag--Solitar relation, establishing equivalences between certain classes of groups and characterizing knot groups with specific torsion properties.

Contribution

It proves that R-groups and ar R-groups coincide for knot groups and characterizes knot groups with generalized torsion of order two.

Findings

R-groups and ar R-groups are equivalent for knot groups.

Knot groups with generalized torsion of order two are characterized.

The paper clarifies the structure of knot groups related to torsion and Baumslag--Solitar relations.

Abstract

Let $G$ be a group. If an equation $x^n = y^n$ in $G$ implies $x = y$ for any elements $x$ and $y$, then $G$ is called an $R$--group. It is completely understood which knot groups are $R$--groups. Fay and Walls introduced $\bar{R}$--group in which the normalizer and the centralizer of an isolator of $\langle x \rangle$ coincide for any non-trivial element $x$. It is known that $\bar{R}$--groups and $R$--groups share many interesting properties and $\bar{R}$--groups are necessarily $R$--groups. However, in general, the converse does not hold. We will prove that these classes are the same for knot groups. In the course of the proof, we will determine knot groups with generalized torsion of order two.