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

Abstract

For a finitely generated group $G$ the Nielsen graph $N_n(G)$, $n\ge \mathrm{rank}(G)$, describes the action of the group $\mathrm{Aut}{F}_n$ 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 $\mathrm{Aut}{F}_n$, $n\ge 4$. We prove nonamenability of Nielsen graphs $N_n(G)$ for all $n\ge \max \{2,\mathrm{rank}(G)\}$ 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.