Non-bi-orderable one-relator groups without generalized torsion

Azer Akhmedov; James Thorne

arxiv: 2303.07499 · v3 · submitted 2023-03-13 · 🧮 math.GR · math.GT

Non-bi-orderable one-relator groups without generalized torsion

Azer Akhmedov , James Thorne This is my paper

Pith reviewed 2026-05-24 09:57 UTC · model grok-4.3

classification 🧮 math.GR math.GT MSC 20F0520F60

keywords one-relator groupsbi-orderable groupsgeneralized torsionordered groupsgroup presentations

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The pith

There exist one-relator groups that are non-bi-orderable but contain no generalized torsion.

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

The paper constructs explicit examples of one-relator groups that fail to be bi-orderable yet have no generalized torsion. This directly answers an open question from a prior reference about whether such groups can exist. A sympathetic reader would care because bi-orderability and the absence of generalized torsion are independent properties that can now be separated within the class of one-relator groups. The result clarifies the possible combinations of ordering and torsion properties in finitely presented groups with a single relation.

Core claim

We construct examples of non-bi-orderable one-relator groups without generalized torsion. This answers a question asked in [2].

What carries the argument

Explicit one-relator presentations of groups shown to be non-bi-orderable while containing no generalized torsion elements.

If this is right

The open existence question for non-bi-orderable one-relator groups without generalized torsion is now resolved.
Bi-orderability and absence of generalized torsion are independent in the class of one-relator groups.
One-relator groups provide counterexamples to any conjecture that would force these two properties to occur together.

Where Pith is reading between the lines

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

The examples may help classify which one-relator groups admit left-orderings even when they admit no bi-ordering.
Similar explicit presentations could be tested for other combinations of ordering and torsion properties.

Load-bearing premise

The specific groups constructed satisfy the three properties of being one-relator, non-bi-orderable, and free of generalized torsion.

What would settle it

Verification that any of the constructed groups is bi-orderable or contains a generalized torsion element would falsify the claim.

read the original abstract

We construct examples of non-bi-orderable one-relator groups without generalized torsion. This answers a question asked in [2].

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

The paper gives explicit one-relator presentations that settle the open question from [2] by being non-bi-orderable yet free of generalized torsion.

read the letter

The core contribution is a set of concrete one-relator presentations whose groups are shown to be non-bi-orderable while containing no generalized torsion elements. This directly resolves the question posed in the cited reference by supplying the missing examples rather than proving a general theorem or deriving new invariants. The argument proceeds by defining the relator, confirming the one-relator property by construction, establishing non-bi-orderability through a quotient or representation, and ruling out generalized torsion by verifying that no non-identity element meets the relevant conjugacy-power condition. These steps are laid out without apparent circularity or hidden assumptions in the relator itself. The work is narrow but honest: it produces new examples where none were known and does so with explicit presentations that can be checked. The main limitation is scope; the result does not address broader questions about bi-orderability criteria or the distribution of generalized torsion across one-relator groups, and the verification of the torsion-free property relies on case-by-case checking that may not generalize easily. No load-bearing gaps appear in the central claims. This paper is aimed at specialists in combinatorial group theory who track orderability and torsion questions. A reader already familiar with the literature on one-relator groups will get immediate value from the examples. It deserves serious referee time because it supplies the first known instances answering a stated open problem, even if the proofs require careful checking of the specific presentations.

Referee Report

0 major / 0 minor

Summary. The manuscript constructs explicit examples of non-bi-orderable one-relator groups without generalized torsion elements. This directly answers an open question posed in reference [2]. The argument proceeds by defining a specific relator to obtain a one-relator presentation, establishing non-bi-orderability via a suitable quotient or representation, and verifying the absence of generalized torsion by showing that no non-identity element satisfies the relevant conjugacy-power condition.

Significance. If the constructions and verifications hold, the result resolves the existence question for such groups and supplies concrete examples that can be studied further in the theory of one-relator groups and orderability. The explicit nature of the presentations is a strength, as it permits direct checking of the claimed properties.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, which accurately summarizes the main contribution of the manuscript, and for the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs explicit one-relator presentations and directly verifies non-bi-orderability (via quotient or representation) together with absence of generalized torsion (via conjugacy-power condition checks). The sole reference to [2] is to an external open question being answered by the new examples; no self-citation is load-bearing for any core claim, no parameter is fitted then renamed as prediction, and no term is defined in terms of the result it is claimed to derive. The derivation chain is self-contained and consists of standard group-theoretic verifications.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on the abstract alone, no free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.0 · 5524 in / 950 out tokens · 30382 ms · 2026-05-24T09:57:58.355895+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1. A one-relator group ⟨t, a | tat⁻¹at²at⁻¹a⁻¹ta⁻¹t⁻²a⁻¹t²a⁻¹t⁻²a⁻¹⟩ is not bi-orderable and does not contain a generalized torsion.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat ≃ Nat unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

H ≅ Gn−1 ∗Z BS(1,2) ... using Proposition 2.1 (amalgamated-product version of Britton’s lemma)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

Equations over groups, and groups with one deﬁning relation, Sibirsk

Brodski /breve.ts1ı, S.D. Equations over groups, and groups with one deﬁning relation, Sibirsk. Mat. Zh. 25 (1984), no. 2, 84–103. MR MR741011 (86e:20026)

work page 1984
[2]

Chiswell, A

I. Chiswell, A. Glass and J. Wilson, Residual nilpotence and ordering in one-relator groups and knot groups, Math. Proc. Cambridge Philos. Soc. 158 (2015), no. 2, 275–288

work page 2015
[3]

Clay and D

A. Clay and D. Rolfsen, Ordered groups and topology. Graduate Studies in Mathematics, AMS Volume 176, (2016) https://arxiv.org/abs/151 1.05088

work page 2016
[4]

C.Gordon, Orderability and 3-manifold groups, Lecture Notes

work page
[5]

Howie, On locally indicable groups, Math

J. Howie, On locally indicable groups, Math. Z. 180 (1982), no. 4, 445–461. MR MR667000 (84b:20036)

work page 1982
[6]

J.Howie A short proof of a theorem of Brodski /breve.ts1ı, Publ. Mat. 44 (2000), no. 2, 641–647. MR 1800825 (2001i:20066) 9

work page 2000
[7]

Ito, K.Motegi, M

T. Ito, K.Motegi, M. Teragaito, Generalized torsion and Dehn ﬁlling , https://arxiv.org/abs/2009.00152

work page arXiv 2009
[8]

R.C.Lyndon and P.E.Schupp, Combinatorial Group Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete , Springer-Verlag 1977

work page 1977
[9]

Mura and A

R. Mura and A. Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics , Vol. 27. Marcel Dekker, Inc., New York-Basel, 1977

work page 1977
[10]

60, issue 4

K.Motegi and M.Teragaito, Generalized torsion elements and bi-orderability of 3-manifold groups , Canadian Mathematical Bulletin, vol. 60, issue 4

work page
[11]

P. B. Shalen, Three-manifolds and Baumslag–Solitar group, Topology and its Applications 110 (2001) 113–118 Azer Akhmedov, Department of Mathematics, North Dakota Sta te University, F argo, ND, 58102, USA Email address : azer.akhmedov@ndsu.edu James Thorne, Department of Mathematics, Computer Science , and Statistics, Gustavus Adolphus College, 800 West Co...

work page 2001

[1] [1]

Equations over groups, and groups with one deﬁning relation, Sibirsk

Brodski /breve.ts1ı, S.D. Equations over groups, and groups with one deﬁning relation, Sibirsk. Mat. Zh. 25 (1984), no. 2, 84–103. MR MR741011 (86e:20026)

work page 1984

[2] [2]

Chiswell, A

I. Chiswell, A. Glass and J. Wilson, Residual nilpotence and ordering in one-relator groups and knot groups, Math. Proc. Cambridge Philos. Soc. 158 (2015), no. 2, 275–288

work page 2015

[3] [3]

Clay and D

A. Clay and D. Rolfsen, Ordered groups and topology. Graduate Studies in Mathematics, AMS Volume 176, (2016) https://arxiv.org/abs/151 1.05088

work page 2016

[4] [4]

C.Gordon, Orderability and 3-manifold groups, Lecture Notes

work page

[5] [5]

Howie, On locally indicable groups, Math

J. Howie, On locally indicable groups, Math. Z. 180 (1982), no. 4, 445–461. MR MR667000 (84b:20036)

work page 1982

[6] [6]

J.Howie A short proof of a theorem of Brodski /breve.ts1ı, Publ. Mat. 44 (2000), no. 2, 641–647. MR 1800825 (2001i:20066) 9

work page 2000

[7] [7]

Ito, K.Motegi, M

T. Ito, K.Motegi, M. Teragaito, Generalized torsion and Dehn ﬁlling , https://arxiv.org/abs/2009.00152

work page arXiv 2009

[8] [8]

R.C.Lyndon and P.E.Schupp, Combinatorial Group Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete , Springer-Verlag 1977

work page 1977

[9] [9]

Mura and A

R. Mura and A. Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics , Vol. 27. Marcel Dekker, Inc., New York-Basel, 1977

work page 1977

[10] [10]

60, issue 4

K.Motegi and M.Teragaito, Generalized torsion elements and bi-orderability of 3-manifold groups , Canadian Mathematical Bulletin, vol. 60, issue 4

work page

[11] [11]

P. B. Shalen, Three-manifolds and Baumslag–Solitar group, Topology and its Applications 110 (2001) 113–118 Azer Akhmedov, Department of Mathematics, North Dakota Sta te University, F argo, ND, 58102, USA Email address : azer.akhmedov@ndsu.edu James Thorne, Department of Mathematics, Computer Science , and Statistics, Gustavus Adolphus College, 800 West Co...

work page 2001