1.2. Approximate groups 17 Example 1.2.6 (Extensions). Let φ : G → H be a surjective group ho- momorphism (thus G is a group extension of H by the kernel ker(φ) of φ). If A is a K-approximate subgroup of H, then φ−1(A) is a K-approximate subgroup of G. One can think of φ−1(A) as an extension of the approximate group A by ker(φ). The classification of approximate groups is of importance in additive combinatorics, and has connections with number theory, geometric group theory, and the theory of expander graphs. One can ask for a quantita- tive classification, in which one has explicit dependence of constants on the approximate group parameter K, or one can settle for a qualitative classifica- tion in which one does not attempt to control this dependence of constants. In this text we will focus on the latter question, as this allows us to bring in qualitative tools such as the Gleason-Yamabe theorem to bear on the problem. In the abelian case when the ambient group G is additive, approximate groups are classified by Freiman’s theorem for abelian groups3 [GrRu2007]. As before, we phrase this theorem in a slightly stilted fashion (and in a qualitative, rather than quantitative, manner) in order to demonstrate its alignment with the general principles stated in the introduction. Theorem 1.2.7 (Freiman’s theorem in an abelian group). Let A be an object with the following properties: (1) (Group-like object) A is a subset of an additive group G. (2) (Weak regularity) A is a K-approximate group. (3) (Discreteness) A is finite. Then there exists a finite subgroup H of G, and a subset P of G/H, with the following properties: (i) (P is close to A/H) π−1(P ) is contained in 4A := A + A + A + A, where π : G → G/H is the quotient map, and |P| K |A|/|H|. (ii) (Nilpotent type structure) P is a symmetric generalised arithmetic progression of rank OK(1) (see Example 1.2.3). Informally, this theorem asserts that in the abelian setting, discrete approximate groups are essentially bounded rank symmetric generalised arithmetic progressions, extended by finite groups (such extensions are also known as coset progressions). The theorem has a simpler conclusion (and is simpler to prove) in the case when G is a torsion-free abelian group (such as Z), since in this case H is trivial. 3The original theorem of Freiman [Fr1973] obtained an analogous classification in the case when G was a torsion-free abelian group, such as the integers Z.

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