# P48: Andrews-Curtis conjecture

- ID: `P48`
- Reference: `andrews-curtis-conjecture`
- Page: https://theoremdb.org/statements/P48
- Record maturity: Reviewed problem with recorded work

## 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.

### Context

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

### Problem setup

- **Definition (A balanced presentation has the same number of generators and relators).** A balanced presentation has the same number of generators and relators.
- **Definition (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.** The conjecture asks whether an algebraically trivial presentation can always be simplified by a small local move set.

### What 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.

## Status

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](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The move set and status were checked against the cited paper and current formal-conjecture records on 2026-07-22.
- Stable variants that allow extra generator-relator pairs are distinct statements.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

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

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `andrews-curtis-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://arxiv.org/abs/math/0302080
   - Also cited at abstract and candidate-presentation discussion
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:math/0302080, checked 2026-07-31; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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 https://arxiv.org/abs/2606.06122
   - preprint; primary source; arXiv:2606.06122v1; checked 2026-08-01
   - Source use: original_summary
   - Proves the original conjecture for the thickenable subclass and gives a stable-move bound without settling the unrestricted conjecture.