Generalized torsion elements and hyperbolic links

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 [Formula: see text], can be generalized to hyperbolic pretzel links of type [Formula: see text]. These give the first examples of hyperbolic links whose link groups admit generalized torsion elements.