Andrews–Curtis conjecture

The Andrews–Curtis conjecture says every balanced presentation of the trivial group can be transformed into the standard presentation by Nielsen moves and conjugations of relators.

Connect

Balanced presentation: A group presentation with the same number of generators as relators. The conjecture concerns the special case where this count matches.
Group presentation: A description of a group by generators and defining relations. The conjecture is a claim about transforming one particular kind of group presentation.
James J. Andrews: An American mathematician known for work in combinatorial group theory and low-dimensional topology. He co-formulated the conjecture with Morton Curtis.
Trivial group presentation: A presentation whose generators and relators define the group with one element. The conjecture predicts a standard reduction for every balanced example.
Andrews–Curtis counterexample: A balanced presentation of the trivial group that cannot be reduced to the standard presentation by Andrews–Curtis moves. No accepted counterexample is known, but finding one would disprove the conjecture.
Andrews–Curtis moves: Operations on a group presentation that change generators or relators while preserving the presented group. These are the permitted steps in the conjectured reduction.
Free group: A group generated by a set with no relations other than those forced by the group axioms. Balanced presentations encode relators as words in a free group.
Morton L. Curtis: An American mathematician whose work included combinatorial group theory and topology. He co-formulated the conjecture with James J. Andrews.
Akbulut–Kirby presentations: A family of balanced presentations of the trivial group proposed in connection with the Andrews–Curtis conjecture. These examples became prominent test cases for the conjecture.
Andrews–Curtis distance: The minimum number of Andrews–Curtis moves needed to connect equivalent presentations, when such a sequence exists. Large or unbounded distances may explain why proposed reductions resist proof.