18 1. Introduction

We will not discuss the proof of Theorem 1.2.7 from [GrRu2007] here,

save to say that it relies heavily on Fourier-analytic methods, and as such,

does not seem to easily extend to a general nonabelian setting. To state the

nonabelian analogue of Theorem 1.2.7, one needs multiplicative analogues of

the concept of a generalised arithmetic progression. An ordinary (symmet-

ric) arithmetic progression {−Nv,...,Nv} has an obvious multiplicative

analogue, namely a (symmetric) geometric progression

{a−N

, . . . ,

aN

} for

some generator a ∈ G. In a similar vein, if one has r commuting generators

a1,...,ar and some dimensions N1,...,Nr 0, one can form a symmetric

generalised geometric progression

(1.4) P

:=

{a1

n1

. . . ar

nr

: |ni| ≤ Ni∀1 ≤ i ≤ r},

which will still be a

3r-approximate

group. However, if the a1,...,ar do not

commute, then the set P defined in (1.4) is not quite the right concept to

use here; for instance, there it is no reason for P to be symmetric. However,

it can be modified as follows:

Definition 1.2.8 (noncommutative progression). Let a1,...,ar be elements

of a (not necessarily abelian) group G = (G, ·), and let N1,...,Nr 0.

We define the noncommutative progression P = P (a1,...,ar; N1,...,Nr)

of rank r with generators a1,...,ar and dimensions N1,...,Nr to be the

collection of all words w composed using the alphabet a1,a1

−1,...,ar,ar −1,

such that for each 1 ≤ i ≤ r, the total number of occurrences of ai and ai

−1

combined in w is at most Ni.

Example 1.2.9. P (a, b;1, 2) consists of the elements

1,a±,b±,b±a±,a±b±,b±2,b±2a±,b±a±b±,a±b±2,

where each occurrence of ± can independently be set to + or −; thus

P (a, b;1, 2) can have as many as 31 elements.

Example 1.2.10. If the a1,...,ar commute, then the noncommutative pro-

gression P (a1,...,ar; N1,...,Nr) simplifies to (1.4).

Exercise 1.2.2. Let G be the discrete Heisenberg group

(1.5) G =

⎛

⎝0

1 Z Z

1

Z⎠

0 0 1

⎞

and let

e1 :=

⎛

⎝0

1 1 0

1

0⎠

0 0 1

⎞

, e2 :=

⎛

⎝0

1 0 0

1

1⎠

0 0 1

⎞

be the two generators of G. Let N ≥ 1 be a suﬃciently large natural number.

Show that the noncommutative progression P (e1,e2; N, N) contains all the