Andrews–Curtis conjecture with stabilizations

Determine whether every balanced presentation of the trivial group can be transformed into a trivial balanced presentation using relator permutations, relator inversion, multiplication of one relator by another, conjugation of a relator by a generator word, and stabilization by adjoining a generator together with a relator equal to that generator.

Background

The Andrews–Curtis conjecture with stabilizations (ACCS) is presented as a consequence of Zeeman’s collapsibility conjecture. It asserts that balanced presentations of the trivial group can be simplified to the trivial balanced presentation through specified elementary operations, including stabilization.

The paper establishes the equivalence of ACCS and ZCC after proving the reduction from arbitrary contractible two-polyhedra to standard ones. Since ZCC remains unresolved, this equivalent algebraic conjecture also remains unresolved.