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 := {an1 1 . . . 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,a−1,...,ar,a−1, 1 r such that for each 1 i r, the total number of occurrences of ai and a−1 i 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
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