Generalized torsions in once punctured torus bundles

TL;DR

This paper investigates the presence of generalized torsions in the fundamental groups of once punctured torus bundles, providing examples that challenge the conjecture linking generalized torsions and bi-orderability in 3-manifold groups.

Contribution

It demonstrates the existence of generalized torsions in certain once punctured torus bundle groups that are not bi-orderable, including a tunnel number two hyperbolic example.

Findings

Identified generalized torsions in specific non-bi-orderable 3-manifold groups

Provided a hyperbolic example with tunnel number two exhibiting generalized torsions

Challenged the conjecture relating generalized torsions to bi-orderability

Abstract

A generalized torsion in a group, an non-trivial element such that some products of its conjugates is the identity. This is an obstruction for a group being bi-orderable. Though it is known that there is a non bi-orderable group without generalized torsions, it is conjectured that 3-manifold groups without generalized torsions are bi-orderable. In this paper, we find generalized torsions in the fundamental groups of once punctured torus bundles which are not bi-orderable. Our result contains a generalized torsion in a tunnel number two hyperbolic once punctured torus bundle.