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Lemma 8 ([Finite-group displacement]). If F:A→ is any map on a finite abelian group, then there are weights H(χ)≥0 such that, for every z∈A , x\|F(x+z)-F(x)\|1 =∑χ∈A |1-χ(z)|2H(χ). The map F need not be injective.

Proposition 7 ([Separation in the finite tree]). For every node v of height h , every x∈A , and every integer t with Qv t , rLv (Gv(x),Gv(x+t wv))≥4-h Lv.

Lemma 2 ([Finite cut representation]). Let S be finite and f:S→ . There are nonnegative finite weights aC , indexed by nonempty proper subsets C⊂S , such that \|f(x)-f(y)\|1 =∑∅≠C S aC|C(x)-C(y)|. Moreover, ∑∅≠C S aC ≤∑x,y ⊂S, …

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Apply Lemma 2 ([Finite cut representation]) to the finite image of F and pull each cut back to an indicator U:A→0,1 . Since an indicator difference has absolute value equal to its squared absolute value, Parseval gives _x|U(x+z)-U(x)| =∑_ χ∈A |U(χ)|^2|1-χ(z)|^2, U(χ)=_x U(x)χ(x). Multiply by the cut weights and add. The coefficient of each multiplier is the nonnegative number H(χ)=∑_C a_C|U_C(χ)|^2 .

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