Heiko Gimperlein
Institut for Matematiske Fag, Universitetsparken 5, DK2100
København Ø
Tel. +45 353 20772, Fax +45 353 20704
Email: gimperlein at math.ku.dk 
Birger Brietzke (B.Sc. thesis, 2011): Paradoxical decompositions and free subgroups
SO(3)paradoxical decompositions of the 2sphere can be constructed using the free group F2 of rank two. Introducing the concept of equidecomposability, we formalize the BanachTarski 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 F2action. 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 finitelygenerated 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 Lipschitzharmonic 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 finitelygenerated 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 NielsenSchreier, 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 2sphere 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 ShalomTao to Gromov's theorem.
