Nielsen equivalence in small cancellation groups

2010/11/26 by Ilya Kapovich, Richard Weidmann, Kapovich, Ilya +1 · 1 citation
Mathematics · #20F #Advanced Operator Algebra Research #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1011.5862

openalex publication_date 2010/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a group given by the presentation [,] where k≥ 2 and where the u_i∈ F(b₁,..., b_k) and w_i∈ F(a₁,..., a_k) are random words. Generically such a group is a small cancellation group and it is clear that (a₁,...,a_k) and (b₁,...,b_k) are generating n-tuples for G. We prove that for generic choices of u₁,..., u_k and v₁,..., v_k the "once-stabilized" tuples (a₁,..., a_k,1) and (b₁,...,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 (a₁,..., a_k) and (b₁,...,b_k) Nielsen equivalent.

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