Probabilistically-like nilpotent groups

TL;DR

This paper develops a model theoretic framework to analyze probabilistically finite nilpotent groups, showing that groups satisfying certain wide-set equations are extensions of nilpotent groups by locally finite groups.

Contribution

It introduces a general model theoretic approach to understand probabilistic properties of finite nilpotent groups, extending previous results to broader classes of groups.

Findings

Groups where certain equations hold on wide sets are extensions of nilpotent groups by locally finite groups.

Applicable to amenable groups and groups in simple theories.

Provides a unifying framework for probabilistic group properties.

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 $[x_1,\ldots,x_k]=1$ 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.