Generalized torsion elements and bi-orderability of 3-manifold groups

TL;DR

This paper investigates the relationship between generalized torsion elements and bi-orderability in 3-manifold groups, proposing a conjecture and verifying it for various classes of 3-manifolds, with implications for understanding their algebraic properties.

Contribution

It conjectures that 3-manifold groups with generalized torsion elements are not bi-orderable and verifies this for specific classes of 3-manifolds, including non-hyperbolic and some hyperbolic cases.

Findings

Conjecture verified for non-hyperbolic, geometric 3-manifolds.

Confirmed for certain infinite families of hyperbolic 3-manifolds.

Proved that generators of Fibonacci groups are generalized torsion elements.

Abstract

It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds, and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group F(2,m) (m>2) is a generalized torsion element.