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.
