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

2022/08/01 by Keisuke Himeno, Himeno, Keisuke, Kimihiko Motegi +3
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2208.00621

Abstract

Let G be a group. If an equation xⁿ = yⁿ 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 R--group in which the normalizer and the centralizer of an isolator of ⟨ x ⟩ coincide for any non-trivial element x. It is known that R--groups and R--groups share many interesting properties and 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.

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