Let G be a group and let X be a finite subset. The isoperimetric method investigates the objective function vertical bar(XB) X vertical bar, defined on the subsets X with vertical bar X vertical bar >= k and vertical bar G (XB)vertical bar >= k, where XB is the product of X by B. In this paper we present all the basic facts about the isoperimetric method. We improve some of our previous results and obtain generalizations and short proofs for several known results. We also give some new applications. Some of the results obtained here will be used in coming papers to improve Kempermann structure Theory.
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