Complicated generalized torsion elements in Seifert fibered spaces with boundary

Abstract

In a group, a nontrivial element is called a generalized torsion element if some non-empty finite product of its conjugates is equal to the identity. There are various examples of torsion-free groups which contain generalized torsion elements. We can define the order of a generalized torsion element as the minimum number of its conjugates required to generate the identity. In previous works, three-manifold groups which contain a generalized torsion element of order two are determined. However, there are few previous studies that examine the order of a generalized torsion element bigger than two. In this paper, we focus on Seifert fibered spaces with boundary, including the torus knot exteriors, and construct concretely generalized torsion elements of order [Formula: see text], [Formula: see text], [Formula: see text] and others in their fundamental groups.