On conjectures of Andrews and Curtis

TL;DR

This paper proves the equivalence of the Andrews--Curtis conjecture with its cyclic version, confirms a satellite conjecture from 1966, and explores the effects of restrictions and stabilizations on the conjecture.

Contribution

It establishes the equivalence between the original and cyclic Andrews--Curtis conjectures and analyzes the impact of restrictions and stabilizations on their validity.

Findings

Proves the equivalence of the original and cyclic conjectures.

Confirms a satellite conjecture from 1966.

Shows restrictions do not affect the conjecture with stabilizations.

Abstract

It is shown that the original Andrews--Curtis conjecture on balanced presentations of the trivial group is equivalent to its "cyclic" version in which, in place of arbitrary conjugations, one can use only cyclic permutations. This, in particular, proves a satellite conjecture of Andrews and Curtis made in 1966. We also consider a more restrictive "cancellative" version of the cyclic Andrews--Curtis conjecture with and without stabilizations and show that the restriction does not change the Andrews--Curtis conjecture when stabilizations are allowed. On the other hand, the restriction makes the conjecture false when stabilizations are not allowed.