Birger Brietzke (B.Sc. thesis, 2011):
Paradoxical decompositions and free subgroups

SO(3)-paradoxical decompositions of the 2-sphere can be constructed using the free group F2 of rank two. Introducing the concept of equidecomposability, we formalize the Banach-Tarski paradox, stating that the unit ball can be duplicated using rotations and translations only.
Similar paradoxical decompositions exist for any group containing F2 and for spaces with free F2-action. On the other hand, a group G, not allowing a paradoxical decomposition, is called amenable. A surprising fact allows to characterize this property in geometric terms.
The Cayley graph of a finitely-generated group G, with generating set S, is the graph whose vertices are the elements of G, where two vertices are connected, if they differ by some element in S. Properties of G encoded in the Cayley graph may now be analyzed using tools from geometric analysis. As key fact we present Kleiner's theorem: The space of Lipschitz-harmonic functions on the Cayley graph is finite dimensional. As a consequence, we outline a proof of Gromov's theorem about special amenable groups - those of polynomial growth:

Theorem: Let G be a finitely-generated group of polynomial growth. Then there exists a subgroup H of finite index which is nilpotent.

Concerning the structure of the thesis: Chapter 1 discusses the basic properties of free groups, including Nielsen-Schreier, stating that every subgroup of a free group is free. The issue of Ch. 2 is to prove the existence of paradoxical decompositions of the 2-sphere and more generally for any open subset of Rn. Amenable groups and their various characterizations are the topic of Ch. 3. Ch. 4 presents the approach of Kleiner and Shalom-Tao to Gromov's theorem.