[#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.
1Context
The conjecture asks whether an algebraically trivial presentation can always be simplified by a small local move set.
2Problem setup
Definition 1 (A balanced presentation has the same number of generators and relators). A balanced presentation has the same number of generators and relators.
Definition 2 (The permitted Nielsen moves invert a relator or multiply one relator by another; conjugation replaces a relator r by w r w^(-1). The permitted Nielsen moves invert a relator or multiply one relator by another; conjugation replaces a relator r by w r w^(-1).
Remark 1. The conjecture asks whether an algebraically trivial presentation can always be simplified by a small local move set.
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
Current 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
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
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
- Whitehead asphericity conjecturegroup theory
- A Tarski monster of exponent fivegroup theory
- A support-three zero divisor over a torsion-free groupgroup theory
How to cite
TheoremDB contributors, “Andrews-Curtis conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/andrews-curtis-conjectureThis page as plain text: andrews-curtis-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
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.