On Rourke’s Extension of Group Presentations and a Cyclic Version of the Andrews–curtis Conjecture

In 1979, Rourke proposed to extend the set of cyclically reduced defining words of a group presentation 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 if the group, defined by the presentation P, is trivial. We generalize this result by proving it for every group presentation 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.