Andrews-Curtis and Nielsen equivalence relations on some infinite groups

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 $\mathrm{Aut}{F}_n$ and of $F_n$ on normally generating $n$-tuples. The equivalence corresponding to the action of $\mathrm{Aut}{F}_n$ 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.

arXiv commentAccepted to Journal of Group Theory