On transitivity and (non)amenability of Aut(F_n) actions on group presentations

TL;DR

This paper investigates the nonamenability of Nielsen graphs associated with finitely generated groups, revealing new results for indicable and elementary amenable groups, and analyzing specific examples like free Burnside groups.

Contribution

It proves nonamenability of Nielsen graphs for indicable and elementary amenable groups and provides explicit descriptions for certain relatively free groups.

Findings

Nielsen graphs are nonamenable for indicable groups.

Nonamenability holds for large n in elementary amenable groups.

Explicit descriptions of Nielsen graphs for free polynilpotent and Burnside groups.

Abstract

For a finitely generated group $G$ the Nielsen graph $N_n(G)$, $n\geq \operatorname{rank}(G)$, describes the action of the group $\operatorname{Aut}F_n$ of automorphisms of the free group $F_n$ on generating $n$-tuples of G by elementary Nielsen moves. The question of (non)amenability of Nielsen graphs is of particular interest in relation with the open question about Property $(T)$ for $\operatorname{Aut}F_n$, $n\geq 4$. We prove nonamenability of Nielsen graphs $N_n(G)$ for all $n\ge \max\{2,\operatorname{rank}(G)\}$ when $G$ is indicable, and for $n$ big enough when $G$ is elementary amenable. We give an explicit description of $N_d(G)$ for relatively free (in some variety) groups of rank $d$ and discuss their connectedness and nonamenability. Examples considered include free polynilpotent groups and free Burnside groups.