Pseudofinite groups with NIP theory and definability in finite simple groups

TL;DR

This paper investigates the structure of pseudofinite groups with NIP theory, showing conditions under which they are soluble-by-finite, and explores model-theoretic insights into finite simple groups and word maps.

Contribution

It proves that pseudofinite groups with NIP and bounded centraliser chains are soluble-by-finite, extending previous results for stable groups and analyzing definability in finite simple groups.

Findings

Pseudofinite NIP groups with bounded centraliser chains are soluble-by-finite.

Existence of NIP pseudofinite groups that are not soluble-by-finite.

Bound on the soluble radical in classes of finite groups with NIP ultraproducts.

Abstract

We show that any pseudofinite group with NIP theory and with a finite upper bound on the length of chains of centralisers is soluble-by-finite. In particular, any NIP rosy pseudofinite group is soluble-by-finite. This generalises, and shortens the proof of, an earlier result for stable pseudofinite groups. An example is given of an NIP pseudofinite group which is not soluble-by-finite. However, if C is a class of finite groups such that all infinite ultraproducts of members of C have NIP theory, then there is a bound on the index of the soluble radical of any member of C. We also survey some ways in which model theory gives information on families of finite simple groups, particularly concerning products of images of word maps.