Probabilistic nilpotence in infinite groups

The ‘degree of k-step nilpotence’ a finite group G is the proportion tuples (x1,…, xk+1 ∈ Gk+1 for which simple commutator [x1, …, xk+1] equal to identity. In this paper we study versions an infinite G, with degree nilpotence defined by sampling in various natural ways, such as random walk, or Følner sequence if amenable. our first main result show that finitely generated, then positive and only virtually nilpotent (Theorem 1.5). This generalises both earlier second author treating case k = 1 Shalev groups, uses techniques from these results. We also show, using notion polynomial mappings groups developed Leibman others, large extent does not depend on method 1.12). As part argument generalise showing ϕ mapping into torsion-free group, set roots sparse certain sense 5.1). consider where residually but necessarily generated. Here quotients uniformly bounded below, 1.19), answering question Shalev. proof sub-multiplicative respect 1.21), generalising Gallagher.