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By John von Neumann

Measures and integrals

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So the only cases we have to consider are when f is either a non-adjacent transvection of w onto v or partial conjugation of C ⊂ Γ by w. Suppose f is a (non-adjacent) transvection f : v → vw or f : v → wv. Then f (x) has the property that any two copies of w are separated by v and any two copies of w −1 are separated by v −1 . “Shuffling left” can never switch the order of v and w, so this must also be true in the normal form for f (x). e. f (x) = a1 wa2 w−1 a3 w . . where the ai are words which do not use w or w−1 , so shuffling left can only cancel w-pairs, never increase the power to more than 1.

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Thus the derived length of G satisfies log2 (n) ≤ dl(G) ≤ dl(Un ) < log2 (n) + 1, which translates to the first statement of the proposition. The first inequality of the second statement follows from Lemma 19(2). For the second inequality, we use a theorem of Mal’cev [Ma56], which implies that every solvable subgroup H ⊂ GL(n, Z) is virtually isomorphic to a subgroup of Tn (O), the lower triangular matrices over the ring of integers O in some number field. The first commutator subgroup of Tn (O) lies in Un (O), so vdl(H) ≤ dl(Tn (O)) ≤ dl(Un (O)) + 1 = μ(Un ) + 1.

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