Generalized torsion elements in the fundamental groups of once punctured torus bundles

A generalized torsion element in a group is a non-trivial element such that some products of its conjugates are the identity element. This is an obstruction for a group being bi-orderable. Though it is known that there is a non-bi-orderable group without generalized torsion elements, it is conjectured that 3-manifold groups without generalized torsion elements are bi-orderable. In this paper, we find generalized torsion elements in the fundamental groups of once punctured torus bundles which are not bi-orderable, and this implies that the conjecture is true for the fundamental groups of once punctured torus bundles. Our result contains examples of generalized torsion elements in the fundamental groups of tunnel number two hyperbolic once punctured torus bundles. (c) 2024 Elsevier B.V. All rights reserved.