Algebra Seminar: talk by Steffen Kionke

On this Wednesday, the 16th of November 2022, Steffen Kionke (University of Hagen) has virtually visited the Charlotte Scott Centre for Algebra and gave us a talk. The seminar was attended by members of the University of Lincoln and of the Lund University (Sweden).

Title: Upper bounds for the first L2-Betti number of groups

Abstract: Over the last 30 years L2-Betti numbers have become a major tool in the investigation of infinite groups with intriguing applications. After a short introduction to L2-invariants, we take a closer look at the first L2-Betti number. We 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.