In mathematics, the Andrews–Curtis conjecture states that every balanced

presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...

(i.e. a presentation with the same number of generators and relations) of the

trivial group In mathematics, a trivial group or zero group is a group that consists of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usu ...

can be transformed into a trivial presentation by a sequence of

Nielsen transformation In mathematics, especially in the area of modern algebra known as combinatorial group theory, Nielsen transformations are certain automorphisms of a free group which are a non-commutative analogue of row reduction and one of the main tools used i ...

s on the relators together with conjugations of relators, named after

James J. Andrews James J. Andrews (c. 1829 – June 7, 1862) was a Kentucky civilian who worked for the Union Army during the early years of the American Civil War. He led a daring raid behind enemy lines on the Western and Atlantic Railroad, known as the ...

and Morton L. Curtis who proposed it in 1965. It is difficult to verify whether the conjecture holds for a given balanced presentation or not. It is widely believed that the Andrews–Curtis conjecture is false. While there are no counterexamples known, there are numerous potential counterexamples. It is known that the Zeeman conjecture on collapsibility implies the Andrews–Curtis conjecture.

References

* * Combinatorial group theory Conjectures Unsolved problems in mathematics {{algebra-stub