A brief introduction to approximate groups

Summary

This introduction to approximate groups highlights their connection with super-strong approximation, the Freiman inverse problem, the Helfgott–Lindenstrauss conjecture, and the classification of approximate subgroups of simple algebraic groups over finite fields.

A finite subgroup of a group G is a finite subset stable under product. It is tempting to investigate what happens if we consider finite subsets that are almost stable under products. Are they close to genuine subgroups in some meaningful sense? Before answering this question, we need to make precise what we mean by “almost stable”. The subject of approximate groups first attempts to do just that, and then tackles the more general problem of classifying such objects.

The formal definition of an approximate group given in Definition 1.2 below was introduced by T. Tao [2008] and was in part motivated by its use in the groundbreaking work of Bourgain and Gamburd [2008b] on superstrong approximation for Zariski dense subgroups of SL(2, ℤ). However the origins of the concept can be traced much earlier and people have been studying approximate groups much before they were even defined. A significant part of additive number theory as it developed in the last fifty years, and in particular the study of sets of integers with small doubling (e.g., Freiman’s theorem; see Theorem 1.9 below), was precisely about understanding abelian approximate groups. Similarly the sum-product phenomenon (see Theorem 2.5), which played a key role in superstrong approximation especially in the early developments of the subject (e.g., in [Helfgott 2008]), is equivalent to classifying approximate subgroups of the affine group {x ↦ ax + b}.

This article intends to give a brief introduction to this subject and present some of its recent developments. Worthwhile expository readings on the same topics include [Green 2009; 2012; Breuillard et al. 2013b; Pyber and Szabó 2014].