Math Conjectures

Andrews–Curtis Conjecture

OPENMajorConjectureProposed 1965 · Full conjecture

Canonical statement

For every integer n≥1n\ge1, let Fn=⟨x1,…,xn⟩F_n=\langle x_1,\ldots,x_n\rangle be the free group. If r=(r1,…,rn)∈Fnnr=(r_1,\ldots,r_n)\in F_n^n has normal closure FnF_n, then rr can be transformed to (x1,…,xn)(x_1,\ldots,x_n) by finitely many moves
ri↦ri−1,ri↦rirj (i≠j),ri↦wriw−1 (w∈Fn), r_i\mapsto r_i^{-1},\qquad r_i\mapsto r_i r_j\ (i\ne j),\qquad r_i\mapsto w r_iw^{-1}\ (w\in F_n),
their inverses, and permutations of the entries.
View source LaTeX
For every integer \(n\ge1\), let \(F_n=\langle x_1,\ldots,x_n\rangle\) be the free group. If \(r=(r_1,\ldots,r_n)\in F_n^n\) has normal closure \(F_n\), then \(r\) can be transformed to \((x_1,\ldots,x_n)\) by finitely many moves
\[
  r_i\mapsto r_i^{-1},\qquad
  r_i\mapsto r_i r_j\ (i\ne j),\qquad
  r_i\mapsto w r_iw^{-1}\ (w\in F_n),
\] their inverses, and permutations of the entries.
ALG-GROUP-014:[AndrewsCurtis1965][BurnsMacedonska1993][ShehperEtAl2026AC]
Algebra and representation theoryGeometry and topology

Notes

The Andrews–Curtis conjecture concerns balanced generating systems of free groups. Given a tuple r=(r1,…,rn)r=(r_1,\ldots,r_n) in the free group FnF_n whose normal closure is all of FnF_n, the conjecture asserts that rr can be carried to the standard basis (x1,…,xn)(x_1,\ldots,x_n) by elementary moves: inverting an entry, multiplying one entry by another, conjugating an entry, and permuting the entries. It was raised by Andrews and Curtis in 1965 in a paper on free groups and handlebodies, the moves mirroring manipulations of presentations arising in low-dimensional topology [AndrewsCurtis1965].

Attention has focused on balanced presentations of the trivial group, which supply natural potential counterexamples [BurnsMacedonska1993]. Extensive computation, including recent classifier-based approaches [ShehperEtAl2026AC], has succeeded in trivializing many long-standing candidate presentations; but such searches cannot certify all normally generating tuples, and no invariant is known that could exhibit a counterexample.

The conjecture therefore remains open in both directions: there is neither a general proof nor an obstruction capable of ruling out AC-triviality for any specific tuple.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.