Probabilistic nilpotence in infinite groups

Armando Martino, Matthew Tointon, Motiejus Valiunas, Enric Ventura

Published 2018-05-29Version 1

In our first main result, we show that if G is a finitely generated group then, for certain sequences P_n of probability measures on G, if limsup P_n([x_1,...,x_k]=1) > 0 then G is virtually (k-1)-step nilpotent. This includes the cases in which P_n is the distribution of the nth step of a symmetric random walk, or in which G is amenable and (P_n) is an almost-invariant sequence of probability measures. This generalises both an earlier result of the second author treating the case k=2, and a result of Shalev for finite groups, and uses techniques from both of those earlier results. We also show, using the notion of polynomial mappings of groups developed by Leibman and others, that P_n([x_1,...,x_k]=1) actually converges to a limit that does not depend on the sequence of probability measures. As part of our argument we generalise a result of Leibman by showing that if f is a polynomial mapping into a torsion-free nilpotent group then the set of roots of f is sparse in a certain sense. In our second main result we consider the case where G is residually finite, but not necessarily finitely generated, and (H_n) is a sequence of finite-index normal subgroups with trivial intersection. Here, writing P_n for the uniform probability measure on G/H_n, we again show that if limsup P_n([x_1,...,x_k]=1) > 0 then G is virtually (k-1)-step nilpotent, answering a question of Shalev. As part of our proof we show that, with respect to the uniform probability measure on a finite group, P_n([x_1,...,x_k]=1) is sub-multiplicative with respect to quotients, generalising a result of Gallagher.