Nielsen equivalence in small cancellation groups

TL;DR

This paper demonstrates that in certain small cancellation groups, generic generating tuples are not Nielsen equivalent even after one stabilization, providing a counterexample to a Wiegold-type conjecture in hyperbolic groups.

Contribution

It shows that for generic small cancellation groups, one stabilization is insufficient to make certain generating tuples Nielsen equivalent, challenging existing conjectures.

Findings

Generic groups are small cancellation groups.

Single stabilization does not achieve Nielsen equivalence.

Counterexample to a Wiegold-type conjecture in hyperbolic groups.

Abstract

Let $G$ be a group given by the presentation [<a_1,...,a_k,b_1,... b_k\,| a_i=u_i(\bar b), b_i=v_i(\bar a) \hbox{for} 1\le i\le k>,] where $k\ge 2$ and where the $u_i\in F(b_1,..., b_k)$ and $w_i\in F(a_1,..., a_k)$ are random words. Generically such a group is a small cancellation group and it is clear that $(a_1,...,a_k)$ and $(b_1,...,b_k)$ are generating $n$-tuples for $G$. We prove that for generic choices of $u_1,..., u_k$ and $v_1,..., v_k$ the "once-stabilized" tuples $(a_1,..., a_k,1)$ and $(b_1,...,b_k,1)$ 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 $(a_1,..., a_k)$ and $(b_1,...,b_k)$ Nielsen equivalent.