Some additive applications of the isopermetric approach

Let $G$ be a group and let $X$ be a finite subset. The isoperimetric method investigates the objective function $\mathrm{\mid }(XB)\setminus X\mathrm{\mid }$, defined on the subsets $X$ with $\mathrm{\mid }X\mathrm{\mid }\ge k$ and $\mathrm{\mid }G\setminus (XB)\mathrm{\mid }\ge k$. A subset with minimal where this objective function attains its minimal value is called a $k$–fragment. In this paper we present all the basic facts about the isoperimetric method. We improve some of our previous results and obtaingeneralizations 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.