Pseudofinite structures and simplicity

TL;DR

This paper investigates how pseudofinite dimension in ultraproducts of finite structures relates to model-theoretic properties like simplicity, stability, and forking, with applications to finite groups and algebraic regularity.

Contribution

It identifies conditions linking pseudofinite dimension to simplicity and stability, and explores examples including vector spaces and groups, extending Tao's algebraic regularity lemma.

Findings

Pseudofinite dimension conditions imply simplicity or supersimplicity.

Drop in pseudofinite dimension corresponds to forking.

Measure-theoretic conditions relate to local stability.

Abstract

We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are made to products of sets in finite groups, in particular to word maps, and a generalization of Tao's algebraic regularity lemma is noted.