1.2. Approximate groups 19

group elements

⎛

⎝0

1 a c

1

b⎠

0 0 1

⎞

of G with |a|, |b| ≤ δN and |c| ≤ δN

2

for a

suﬃciently small absolute constant δ 0; conversely, show that all elements

of P (e1,e2; N, N) are of the form

⎛

⎝0

1 a c

1

b⎠

0 0 1

⎞

with |a|, |b| ≤ CN and |c| ≤

CN

2

for some suﬃciently large absolute constant C 0. Thus, informally,

we have

P (e1,e2; N, N) =

⎛

⎝0

1 O(N) O(N

2)

1 O(N)

0 0 1

⎞

⎠

.

It is clear that noncommutative progressions P (a1,...,ar; N1,...,Nr)

are symmetric and contain the identity. However, if the a1,...,ar do not

have any commutative properties, then the size of these progressions can

grow exponentially in N1,...,Nr and will not be approximate groups with

any reasonable parameter K. However, the situation changes when the

a1,...,ar generate a nilpotent

group4:

Proposition 1.2.11. Suppose that a1,...,ar ∈ G generate a nilpotent group

of step s, and suppose that N1,...,Nr are all suﬃciently large depending on

r, s. Then P (a1,...,ar; N1,...,Nr) is an Or,s(1)-approximate group.

We will prove this proposition in Chapter 12.

We can now state the noncommutative analogue of Theorem 1.2.7, proven

in [BrGrTa2011]:

Theorem 1.2.12 (Freiman’s theorem in an arbitrary group). Let A be an

object with the following properties:

(1) (Group-like object) A is a subset of a multiplicative 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 N(H)/H

(where N(H) := {g ∈ G : gH = Hg} is the normaliser of H), with the

following properties:

(i) (P is close to A/H)

π−1(P

) is contained in

A4

:= A·A·A·A, where

π : N(H) → N(H)/H is the quotient map, and |P|

K

|A|/|H|.

(ii) (Nilpotent type structure) P is a noncommutative progression of

rank OK(1), whose generators generate a nilpotent group of step

OK(1).

4A

group G is nilpotent if the lower central series G1

:=

G, G2

:=

[G, G1], G3

:=

[G, G2], . . . ,

etc. eventually becomes trivial.