Nielsen Equivalence in Small Cancellation Groups

Let G be a group given by the presentation 〈a1, . . . , ak , b1, . . . bk | ai = ui(b̄), bi = vi(ā) for 1 ≤ i ≤ k〉, where k ≥ 2 and where the ui ∈ F (b1, . . . , bk) and wi ∈ F (a1, . . . , ak) are random words. Generically such a group is a small cancellation group and it is clear that (a1, . . . , ak) and (b1, . . . , bk) are generating n-tuples for G. We prove for generic choices of u1, . . . , uk and v1, . . . , vk the “once-stabilized” tuples (a1, . . . , ak , 1) and (b1, . . . , bk, 1) are not Nielsen equivalent in G. This provides a counter-example for a Wiegold-type conjecture in the setting of word-hyperbolic groups. We conjecture that in the above construction at least k stabilizations are needed to make the tuples (a1, . . . , ak) and (b1, . . . , bk) Nielsen equivalent.