Examples of left-orderable and non-left-orderable HNN extensions

TL;DR

This paper investigates conditions under which HNN extensions of groups are left-orderable, proving that extensions of torsion-free nilpotent groups are always left-orderable and providing counterexamples for other cases.

Contribution

It establishes that HNN extensions of torsion-free nilpotent groups are left-orderable and constructs examples of non-left-orderable HNN extensions of left-orderable groups.

Findings

HNN extension of torsion-free nilpotent groups is left-orderable

Constructs examples of non-left-orderable HNN extensions of left-orderable groups

Provides criteria for left-orderability in HNN extensions

Abstract

We prove that an HNN extension of a torsion-free nilpotent group is left-orderable. We also construct examples of non-left-orderable HNN extensions of left-orderable groups