Abstract:
Over the last 30 years L2-Betti numbers have become a major tool in
the investigation of infinite groups with intruiging applications.
After a short introduction to L2-invariants, we will take a closer look at the first L2-Betti number. We will explain how the first L2-Betti number can be studied using the geometry of the Cayley graph and present a simple method to extract upper bounds. Some applications are discussed. For instance, the bounds can be used to prove the vanishing of the first L2-Betti number of Burnside groups of large prime exponent.
A remarkable feature of the Cayley graph approach is that it still works for a family of "generalized" first Betti numbers. At the end of the talk we give an outlook to this generalized setting.
