Probabilistically-like nilpotent groups

Abstract

We give a model-theoretic proof of a result of Shalev on probabilistically finite nilpotent groups. More generally, we prove that a 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 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.