HNN extensions of torsion-free nilpotent groups are left-orderable, but HNN extensions of left-orderable groups are not always left-orderable.
Bludov and A.M.W.Glass, Word problems, embeddings, and free p rod- ucts of right-ordered groups with amalgamated subgroup, Proceedings of the London Mathemtical Society , vol
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Examples of left-orderable and non-left-orderable HNN extensions
HNN extensions of torsion-free nilpotent groups are left-orderable, but HNN extensions of left-orderable groups are not always left-orderable.