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.