Balanced presentations of the trivial group on two generators and the Andrews-Curtis conjecture (Q2759634)

It is proved that every balanced presentation of the trivial group with two generators and with total length of the relators \(\leq 12\) is equivalent to the trivial presentation. This means that the tuple of relators \((r_1,r_2)\) can be reduced by a sequence of elementary Nielsen transformations and conjugations to the trivial tuple \((x_1,x_2)\) of generators. In other words the well-known Andrews-Curtis conjecture (ACC) is true in the case considered. Note that a number of presentations proposed before as potential counterexamples to ACC satisfy the assumptions above.NEWLINENEWLINENEWLINEThe proof of the main result is based on computer computations with the software package MAGNUS. Some generic algorithms are applied.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00030].