Expansion in groups: Coarse geometry and analytic aspects

Specialeforsvar ved Mikkel Emil Strunge

Titel: Expansion in group: Coarse geometry and analytic aspects

Abstract: In this thesis we study finitely-generated groups as geometric objects, where the distance between points is defined by some word-length measured on the Cayley graph of the group. We establish connections between analytic properties of groups (amenability, Kazhdan’s property (T) and Haagerup’s property (H)) and large-scale (coarse) geometry properties of certain families of graphs (the so-called box spaces) constructed inside the group. We also introduce and study expander graphs, which are highly connected finite graphs that play an important role in computer science and pure mathematics. While existence of expanders follows from probabilistic arguments, explicit constructions require deep mathematical arguments. We prove Margulis’ result that box spaces of residually finite property (T) groups, such as SL3(Z), are expanders.

Vejleder: Magdalena Musat
Censor:   Bent Ørsted, Aarhus Universitet