On transitivity and (non)amenability of Aut(F_n) actions on group presentations
Aglaia Myropolska, Tatiana Nagnibeda

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 the Nielsen graph , , describes the action of the group of automorphisms of the free group on generating -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 for , . We prove nonamenability of Nielsen graphs for all when is indicable, and for big enough when is elementary amenable. We give an explicit description of for relatively free (in some variety) groups of rank and discuss their connectedness and nonamenability. Examples considered include free polynilpotent groups and free Burnside groups.
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