[#P48] Andrews-Curtis conjecture

Problem. Every balanced presentation \(\langle x_1,\ldots,x_n\mid r_1,\ldots,r_n\rangle\) of the trivial group can be transformed into \(\langle x_1,\ldots,x_n\mid x_1,\ldots,x_n\rangle\) by Andrews-Curtis moves.

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1Context

The conjecture asks whether an algebraically trivial presentation can always be simplified by a small local move set.

2Remarks

Remark 1. A balanced presentation has the same number of generators and relators.

Remark 2. The permitted Nielsen moves invert a relator or multiply one relator by another; conjugation replaces a relator r by w r w^(-1).

3What counts as a solution

Prove that every balanced presentation of the trivial group admits a finite sequence of allowed moves to the standard presentation, or give a balanced trivial-group presentation and prove that no such sequence exists.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Lackenby proves the original conjecture for thickenable balanced presentations and gives an explicit stable-move bound for that class. The unrestricted conjecture remains open. Exact unresolved remainder: Prove that every balanced presentation of the trivial group is Andrews-Curtis equivalent to the standard presentation, or give a balanced presentation of the trivial group and prove that no allowed move sequence reaches the standard presentation.^[1][2]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Lackenby proves the original conjecture for thickenable balanced presentations and gives an explicit stable-move bound for that class. The unrestricted conjecture remains open. Exact unresolved remainder: Prove that every balanced presentation of the trivial group is Andrews-Curtis equivalent to the standard presentation, or give a balanced presentation of the trivial group and prove that no allowed move sequence reaches the standard presentation.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Lackenby proves the original conjecture for thickenable balanced presentations and gives an explicit stable-move bound for that class. The unrestricted conjecture remains open. Exact unresolved remainder: Prove that every balanced presentation of the trivial group is Andrews-Curtis equivalent to the standard presentation, or give a balanced presentation of the trivial group and prove that no allowed move sequence reaches the standard presentation.^[1][2] Relevance to this problem For Andrews-Curtis conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Lackenby proves the original conjecture for thickenable balanced presentations and gives an explicit stable-move bound for that class. The unrestricted conjecture remains open. Exact unresolved remainder: Prove that every balanced presentation of the trivial group is Andrews-Curtis equivalent to the standard presentation, or give a balanced presentation of the trivial group and prove that no allowed move sequence reaches the standard presentation. Open full record →
Prove that every balanced presentation of the trivial group admits a finite sequence of allowed moves to the standard presentation, or give a balanced trivial-group presentation and prove that no such sequence exists.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove that every balanced presentation of the trivial group admits a finite sequence of allowed moves to the standard presentation, or give a balanced trivial-group presentation and prove that no such sequence exists.^[1] Relevance to this problem For Andrews-Curtis conjecture, record andrews-curtis-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove that every balanced presentation of the trivial group admits a finite sequence of allowed moves to the standard presentation, or give a balanced trivial-group presentation and prove that no such sequence exists. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove that every balanced presentation of the trivial group admits a finite sequence of allowed moves to the standard presentation, or give a balanced trivial-group presentation and prove that no such sequence exists. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited paper treats the Andrews-Curtis assertion as a conjecture with unresolved candidate counterexamples, and current public status was checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Stable variants that allow extra generator-relator pairs are distinct statements.

Computational notes

Searches can find long move sequences or resist bounded searches; failure within a search radius does not prove inequivalence.

2See also

A support-three zero divisor over a torsion-free groupgroup theory
A Tarski monster of exponent fivegroup theory
Kaplansky zero-divisor conjecturegroup theory

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Plain text

“Andrews-Curtis conjecture.” TheoremDB. P48. Problem statement; statement text SHA-256 131a45a5e3fc427d371e843f0c67d82111e63f099cd8918e146eb728ff9799e6. https://theoremdb.org/statement/?ref=P48

BibTeX

@misc{theoremdb-problem-131a45a5e3fc427d371e843f0c67d82111e63f099cd8918e146eb728ff9799e6,
  title = {{Andrews-Curtis conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 131a45a5e3fc427d371e843f0c67d82111e63f099cd8918e146eb728ff9799e6},
  url = {https://theoremdb.org/statement/?ref=P48}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Alexei D. Myasnikov, Alexei G. Myasnikov, and Vladimir Shpilrain, “On the Andrews-Curtis equivalence”. Contemp. Math., Amer. Math. Soc. 296 (2002), 183-198. arXiv:math/0302080 (2003). Alexei D. Myasnikov, Alexei G. Myasnikov, and Vladimir Shpilrain, arXiv:math/0302080, abstract. ↗preprint · primary source · arXiv:math/0302080, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited paper treats the Andrews-Curtis assertion as a conjecture with unresolved candidate counterexamples, and current public status was checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at abstract and candidate-presentation discussion.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Defines the equivalence problem and records the unresolved candidate boundary.Source named by the research packet.
Marc Lackenby, “The stable Andrews-Curtis conjecture and thickenable presentations of the trivial group”. arXiv:2606.06122 (2026). abstract and principal theorems on thickenable presentations. ↗preprint · primary source · arXiv:2606.06122v1 · checked 2026-08-01Source use: original summary.Proves the original conjecture for the thickenable subclass and gives a stable-move bound without settling the unrestricted conjecture.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P48] Andrews-Curtis conjecture

1Context

2Remarks

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1Lean verification

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1References

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1Context

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Computational notes

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Background

Definitions

What counts as a solution

What's known

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Related work

Record & provenance