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

Azer Akhmedov; Cody Martin

arxiv: 2301.00052 · v3 · submitted 2022-12-30 · 🧮 math.GR · math.GT

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

Azer Akhmedov , Cody Martin This is my paper

Pith reviewed 2026-05-24 09:28 UTC · model grok-4.3

classification 🧮 math.GR math.GT

keywords left-orderable groupsHNN extensionstorsion-free nilpotent groupsgroup orderabilitynon-left-orderable examplesHNN extension of nilpotent groups

0 comments

The pith

An HNN extension of a torsion-free nilpotent group is left-orderable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that whenever the base group of an HNN extension is torsion-free and nilpotent, the resulting group admits a left-order. It also exhibits concrete HNN extensions of left-orderable groups that fail to be left-orderable. A reader cares because left-orderability determines whether a group can act faithfully on the real line by order-preserving maps, a property preserved under some but not all group constructions. The result therefore marks a sharp boundary: the nilpotency and torsion-freeness hypotheses are both necessary for the preservation to hold in this setting.

Core claim

The authors prove that an HNN extension of any torsion-free nilpotent group is left-orderable. They further construct explicit examples of HNN extensions whose base groups are left-orderable yet the full groups are not left-orderable.

What carries the argument

The HNN extension of a base group by a stable letter that conjugates one subgroup to another, together with the extension of a left-order from the base using its nilpotency and torsion-freeness.

If this is right

Left-orderability passes from any torsion-free nilpotent group to all of its HNN extensions.
Left-orderability is not preserved under HNN extensions when the base group is an arbitrary left-orderable group.
The class of left-orderable groups is not closed under HNN extensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Nilpotency supplies an inductive structure that lets an order on the base be lifted step-by-step across the stable letter.
The same lifting technique may or may not apply to other solvable or polycyclic groups; concrete small examples could be checked by hand.
The counterexamples show that left-orderability can be destroyed by adjoining a single stable letter even when the base acts on the line.

Load-bearing premise

The base group must be both torsion-free and nilpotent; without both conditions the order on the base need not extend to the HNN extension.

What would settle it

An explicit HNN extension of a torsion-free nilpotent group with no left-order, or a proof that no such order exists for some concrete base.

read the original 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

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

HNN extensions of torsion-free nilpotent groups are left-orderable, with counterexamples showing the condition is necessary.

read the letter

The main takeaway is that the paper proves every HNN extension of a torsion-free nilpotent group is left-orderable, while also constructing examples of left-orderable groups whose HNN extensions fail to be left-orderable. This cleanly separates the cases and shows the nilpotency hypothesis is essential rather than incidental. The positive result and the explicit counterexamples both appear new based on the abstract and stress-test. The paper does well by giving both the preservation theorem and the sharpness examples in one place, which makes the distinction useful for anyone testing similar questions about orderability under amalgamated constructions. The counterexamples are likely the part that will see the most immediate use. The soft spots are limited. The abstract supplies no proof outline, so the details of how the left-order extends across the stable letter must be checked in the full text; if that step uses the nilpotency in an essential way without hidden gaps, the claim holds. The counterexample constructions also need direct verification, but nothing in the stress-test flags an inconsistency or unsupported step. The citation pattern looks standard and appropriate for the area. This paper is for specialists in geometric group theory who work on left-orderable groups and their behavior under HNN extensions. A reader already interested in orderability questions will find the concrete results and the precise condition worth having. It deserves a serious referee because the claims are specific, the counterexamples are checkable, and the overall question is natural inside the field. I recommend sending it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that every HNN extension of a torsion-free nilpotent group is left-orderable. It also constructs explicit examples of HNN extensions of left-orderable (but non-nilpotent) groups that fail to be left-orderable.

Significance. If the claims hold, the result supplies a positive theorem on preservation of left-orderability under HNN extensions when the base group is torsion-free nilpotent, together with counterexamples establishing that nilpotency is essential. This clarifies the boundary between orderable and non-orderable HNN extensions and supplies concrete constructions that can be checked directly.

minor comments (2)

[Abstract] The abstract announces the two results but supplies no indication of the proof method or the form of the counterexamples; a one-sentence outline of each argument would improve readability.
[§1] Notation for the associated subgroups and the stable letter is introduced without an explicit reference to the standard HNN presentation; adding a displayed equation for the HNN extension at the first use would remove ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No circularity: direct proof and explicit counterexamples

full rationale

The paper states and proves a theorem that every HNN extension of a torsion-free nilpotent group is left-orderable, using the nilpotency and torsion-freeness hypotheses to extend a left-order from the base group across the stable letter. It separately constructs explicit examples of left-orderable groups whose HNN extensions fail to be left-orderable, showing the nilpotency condition is essential. No equations, definitions, or claims reduce by construction to fitted inputs or prior self-citations; the argument is self-contained group-theoretic reasoning with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on the standard definitions of HNN extensions and left-orderability from the group-theory literature; no new entities or fitted constants are introduced in the abstract.

axioms (2)

standard math Standard axioms of groups, subgroups, and isomorphisms used to define HNN extensions
Invoked implicitly in the statement of the main theorem.
standard math Definition of left-orderability via a total left-invariant order on the group elements
Central to both the positive and negative statements.

pith-pipeline@v0.9.0 · 5538 in / 1245 out tokens · 38232 ms · 2026-05-24T09:28:20.342252+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Button, Topics in inﬁnite groups, Lecture Notes

J. Button, Topics in inﬁnite groups, Lecture Notes

work page
[2]

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

V.V. 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. 99, issue 3, (2009), 585-608

work page 2009
[3]

Lyndon and P

R. Lyndon and P. Schupp. Combinatorial Group Theory, Volume 8 9 of Ergeb- nisse der Mathematik und ihrer Grenzgebiete, Springer-Verlaq, 19 77

work page
[4]

Navas, Groups of Circle Diﬀeomorphisms

A. Navas, Groups of Circle Diﬀeomorphisms. The University of Chic ago Press, 2011

work page 2011
[5]

M.S.Raghunathan, Discrete Subgroups of Lie Groups, Springer B erlin Heidel- berg, Nov 16, 1972 Azer Akhmedov, Department of Mathematics, North Dakota Sta te University, F argo, ND, 58102, USA Email address : azer.akhmedov@ndsu.edu Cody Martin, Department of Mathematics, North Dakota State University, F argo, ND, 58102, USA Email address : cody.martin@ndsu.edu

work page 1972

[1] [1]

Button, Topics in inﬁnite groups, Lecture Notes

J. Button, Topics in inﬁnite groups, Lecture Notes

work page

[2] [2]

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

V.V. 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. 99, issue 3, (2009), 585-608

work page 2009

[3] [3]

Lyndon and P

R. Lyndon and P. Schupp. Combinatorial Group Theory, Volume 8 9 of Ergeb- nisse der Mathematik und ihrer Grenzgebiete, Springer-Verlaq, 19 77

work page

[4] [4]

Navas, Groups of Circle Diﬀeomorphisms

A. Navas, Groups of Circle Diﬀeomorphisms. The University of Chic ago Press, 2011

work page 2011

[5] [5]

M.S.Raghunathan, Discrete Subgroups of Lie Groups, Springer B erlin Heidel- berg, Nov 16, 1972 Azer Akhmedov, Department of Mathematics, North Dakota Sta te University, F argo, ND, 58102, USA Email address : azer.akhmedov@ndsu.edu Cody Martin, Department of Mathematics, North Dakota State University, F argo, ND, 58102, USA Email address : cody.martin@ndsu.edu

work page 1972