Andrews-Curtis and Nielsen equivalence relations on some infinite groups

TL;DR

This paper investigates Andrews-Curtis and Nielsen equivalence relations within certain infinite groups, including nilpotent and Grigorchuk groups, to analyze their properties and implications for the conjecture.

Contribution

It extends the study of these equivalences to finitely generated groups with all maximal subgroups normal, providing new insights into their structure and potential counter-examples.

Findings

Analysis of equivalences in nilpotent groups

Extension to Grigorchuk groups

Implications for the Andrews-Curtis conjecture

Abstract

The Andrews-Curtis conjecture asserts that, for a free group $F_n$ of rank $n$ and a free basis $(x_1,...,x_n)$, any normally generating tuple $(y_1,...,y_n)$ is Andrews-Curtis equivalent to $(x_1,...,x_n)$. This equivalence corresponds to the actions of $\operatorname{Aut}F_n$ and of $F_n$ on normally generating $n$-tuples. The equivalence corresponding to the action of $\operatorname{Aut}F_n$ on generating $n$-tuples is called Nielsen equivalence. The conjecture for arbitrary finitely generated group has its own importance to analyse potential counter-examples to the original conjecture. We study the Andrews-Curtis and Nielsen equivalence in the class of finitely generated groups for which every maximal subgroup is normal, including nilpotent groups and Grigorchuk groups.