On Rourke’s extension of group presentations and a cyclic version of the Andrews–Curtis conjecture
S. Ivanov · Proceedings of the American Mathematical Society · 2005
In 1979, Rourke proposed to extend the set of cyclically reduced defining words of a group presentation P \mathcal P by using operations of cyclic permutation, inversion and taking double products. He proved that iterations of these operations yield all cyclically reduced words of the normal closure of defining words of P \mathcal P if the group, defined by the presentation P \mathcal P , is trivial. We generalize this result by proving it for every group presentation P \mathcal P with an obvious exception. We also introduce a new, “cyclic", version of the Andrews–Curtis conjecture and show that the original Andrews–Curtis conjecture with stabilizations is equivalent to its cyclic version.