Stable Andrews–Curtis conjecture

A balanced group presentation has the same number of generators and relators. A stable Andrews–Curtis transformation is one of the standard Andrews–Curtis moves together with stabilization by adding or removing a generator-relator pair. The trivial presentation is a presentation of the trivial group with generators and relators freely and normally equivalent to the trivial ones.

Stable Andrews–Curtis conjecture. Every balanced group presentation for the trivial group may be changed to the trivial presentation by a finite sequence of stable Andrews–Curtis transformations.

The conjecture is a proposed method for recognizing the trivial group from balanced presentations and is relevant to the study of potential counterexamples to the smooth four-dimensional Poincaré conjecture. The supplied source states that it is generally believed to be false.