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

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.