Generalized torsion elements and hyperbolic links

TL;DR

This paper investigates generalized torsion elements in the fundamental groups of certain hyperbolic links, providing the first examples of such links with these elements, which obstruct bi-orderability.

Contribution

It demonstrates that specific hyperbolic links, including the Whitehead sister link and its generalizations, admit generalized torsion elements in their groups.

Findings

Weeks manifold admits generalized torsion elements.

Whitehead sister link's group admits generalized torsion.

Hyperbolic pretzel links of type (-2,3,2n) also admit these elements.

Abstract

In a group, a generalized torsion element is a non-identity element whose some non-empty finite product of its conjugates yields the identity. Such an element is an obstruction for a group to be bi-orderable. We show that the Weeks manifold, the figure-eight sister manifold, and the complement of Whitehead sister link admit generalized torsion elements in their fundamental groups. In particular, the Whitehead sister link, which is the pretzel link of type (-2,3,8), can be generalized to hyperbolic pretzel links of type (-2,3,2n), n>3. These give the first examples of hyperbolic links whose link groups admit generalized torsion elements.