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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). 4 A group G is nilpotent if the lower central series G1 := G, G2 := [G, G1], G3 := [G, G2], . . . , etc. eventually becomes trivial.