Probabilistically-like nilpotent groups

Abstract

The main goal of the paper is to present a general model theoretic framework to understand a result of Shalev on probabilistically finite nilpotent groups. We prove that a suitable group where the equation [x_1,...,x_k]=1 holds on a wide set, in a model theoretic sense, is an extension of a nilpotent group of class less than k by a uniformly locally finite group. In particular, this result applies to amenable groups, as well as to suitable model-theoretic families of definable groups such as groups in simple theories and groups with finitely satisfiable generics.