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

Presentation of a group

In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...

 of the trivial group

Trivial group

In mathematics, a trivial group is a group consisting 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 so it usually denoted as such, 0, 1 or e depending on the context...

 can be transformed into a trivial presentation by a sequence of Nielsen transformation

Nielsen transformation

In mathematics, especially in the area of abstract algebra known as combinatorial group theory, Nielsen transformations, named after Jakob Nielsen, are certain automorphisms of a free group which are a non-commutative analogue of row reduction and one of the main tools used in studying free groups,...

s on the relators together with conjugations of relators, named after James J. Andrews

James J. Andrews (mathematician)

James J. Andrews was an American mathematician, a professor of mathematics at Florida State University who specialized in knot theory, topology, and group theory....

 and Morton L. Curtis

Morton L. Curtis

Morton Landers Curtis was an American mathematician, an expert on group theory and the W. L. Moody, Jr. Professor of Mathematics at Rice University....

 who proposed it in 1965. It is difficult to verify whether the conjecture holds for a given balanced presentation or not.

Although it is believed that the Andrews–Curtis conjecture is false, there are no counter-examples known, nor are there many good ideas for possible counter-examples. It is known that the Zeeman conjecture on collapsibility implies the Andrews–Curtis conjecture.