Banach Tarski Paradox and Amenability

The main objective of this bachelor thesis is to prove Banach-Tarski theorem. The theorem states that a ball in a 3-dimensional space can be split into finitely many pieces that can be rearranged to form two balls, each of the same size as the first one. The concept of amenability, which underlies the paradox, will be explained and characterized as well. We will also classify some groups in terms of amenability. Proving that groups in certain classes are all amenable and those in others classes are not is the approach that we will take to address this issue.