Nielsen equivalence in small cancellation groups

الأصل · EN

Let G be a group given by the presentation [<a₁,...,aₖ,b₁,... bₖ| aᵢ=uᵢ(b), bᵢ=vᵢ(a) 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 that 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.