We show that a non-empty subset of an abelian group with a small edge boundary must be large; in particular, if A and S are finite, non-empty subsets of an abelian group such that S is independent, and the edge boundary of A with respect to S does not exceed (1 − γ)|S||A| with a real γ ∈ (0, 1], then |A| ≥ 4 (1−1/d)γ|S| , where d is the smallest order of an element of S. Here the constant 4 is best possible. As a corollary, we derive an upper bound for the size of the largest independent subset

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... of the set of popular differences of a finite subset of an abelian group. For groups of exponent 2 and 3, our bound translates into a sharp estimate for the additive dimension of the popular difference set. We also prove, as an auxiliary result, the following estimate of possible independent interest: if A ⊆ Z n is a finite, non-empty downset then, denoting by w(a) the number of non-zero components of the vector a ∈ A, we have 1 |A| ∑ a∈A w(a) ≤ 1 2 log 2 |A|.