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 $\mathcal P$ 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 $\mathcal P$ if the group, defined by the presentation $\mathcal P$, is trivial. We generalize this result by proving it for every group presentation $\mathcal P$ 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.