Math 2.0

Group theory and generalizations Infinite groups and combinatorial group theory Conjecture

Andrews–Curtis conjecture

Builds on Presentation of a groupFree groupConjugate elements

Statement

A presentation ⟨x1,…,xn∣r1,…,rn⟩\langle x_1, \ldots, x_n \mid r_1, \ldots, r_n \rangle is balanced if it has as many relators as generators. The Andrews–Curtis moves on such a presentation, with rir_i elements of the free group FF on x1,…,xnx_1, \ldots, x_n, are:

  1. replace some rir_i by ri−1r_i^{-1};
  2. replace some rir_i by rirjr_i r_j for some j≠ij \ne i;
  3. replace some rir_i by grig−1g r_i g^{-1} for some g∈Fg \in F.

The conjecture: every balanced presentation of the trivial group can be turned into the trivial presentation ⟨x1,…,xn∣x1,…,xn⟩\langle x_1, \ldots, x_n \mid x_1, \ldots, x_n \rangle by a finite sequence of Andrews–Curtis moves.

Remarks

Each move keeps the normal closure of the relators, so it keeps the group. A presentation of the trivial group with nn generators has at least nn relators, since the abelianization ℤn\Z^n must become trivial; so the balanced ones are the extreme case. A weaker stable form of the conjecture also allows adding a generator xn+1x_{n+1} with relator xn+1x_{n+1}, and removing such a pair. Andrews and Curtis stated the conjecture in 1965, in connection with the topology of 22-dimensional complexes and 44-dimensional manifolds. It is open; many experts expect it to be false, and several families of balanced presentations of the trivial group are studied as potential counterexamples.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Conjugate elements, concept, no evidence levelConjugate elementsconceptFree group, concept, no evidence levelFree groupconceptPresentation of a group, concept, no evidence levelPresentation of a gro…conceptAndrews–Curtis conjecture, conjecture, no evidence levelAndrews–Curtis conjec…conjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Conjugate elements concept
  2. Free group concept
  3. Presentation of a group concept
  4. Andrews–Curtis conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Andrews%E2%80%93Curtis_conjecture
About this page
TagN298DRYE0W
URIm2://N298DRYE0W
Kindconjecture
WritingW0 stub
MSC20F05, 57M20
StatusThis copy of the site has no build data. Run m2 status N298DRYE0W.
AboutPresentation of a group
UsesPresentation of a group, Free group, Conjugate elements
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Changed2 November 2026

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