Nielsen equivalence in Gupta-Sidki groups

TL;DR

This paper proves that for Gupta-Sidki p-groups with prime p ≥ 3, there are infinitely many Nielsen equivalence classes among generating pairs, revealing complex symmetry structures.

Contribution

It establishes the existence of infinitely many Nielsen equivalence classes in Gupta-Sidki p-groups, a novel result in the study of their generating sets.

Findings

Infinitely many Nielsen classes for generating pairs in G_p

Complex automorphism orbits in Gupta-Sidki groups

Advances understanding of group symmetries

Abstract

For a group $G$ generated by $k$ elements, the Nielsen equivalence classes are defined as orbits of the action of $\operatorname{Aut} F_k$, the automorphism group of the free group of rank $k$, on the set of generating $k$-tuples of $G$. Let $p\geq 3$ be prime and $G_p$ the Gupta-Sidki $p$-group. We prove that there are infinitely many Nielsen equivalence classes on generating pairs of $G_p$.