Some additive applications of the isopermetric approach

TL;DR

This paper explores the isoperimetric method in group theory, focusing on $k$-fragments, providing foundational facts, improvements on previous results, and new applications to advance Kempermann structure theory.

Contribution

It offers a comprehensive overview of the isoperimetric method, improves existing results, and introduces new applications relevant to Kempermann structure theory.

Findings

Characterization of $k$-fragments in groups

Improved bounds and properties of the isoperimetric function

New applications in group structure analysis

Abstract

Let $G$ be a group and let $X$ be a finite subset. The isoperimetric method investigates the objective function $|(XB)\setminus X|$, defined on the subsets $X$ with $|X|\ge k$ and $|G\setminus (XB)|\ge k$. 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.