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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Aglaia Myropolska, Tatiana Nagnibeda, On transitivity and (non)amenability of Aut actions on group presentations. Groups Geom. Dyn. 8 (2014), no. 3, pp. 837–861DOI 10.4171/GGD/250