| Résume | At this point, it’s well known that the Baum—Connes conjecture has various counterexamples to its many forms (the version for groupoids, and the version with coefficient algebras attached and the geometric version). The groups for which the conjecture fails with coefficients are known as “Gromov Monsters”, and they contain expander graphs embedded in their Cayley graphs in some weak metric sense. These groups are produced using probabilistic methods and rely on small cancellation theory to confirm their existence. Many such examples can be made “as hyperbolic as possible” even whilst not being finitely presented -this should be take to mean that they’re direct limits of a sequence of hyperbolic groups that have a strong relationship between the injectivity radius of the connecting maps in the limit diagram and the hyperbolicity of the sequence terms. This class of groups was dubbed “lacunary hyperbolic” by Osin—Ol’Shanskii—Sapir, and is well known to contain a variety of complex and pathological examples of other monster groups. The main aim of the talk is to give an overview of how to use quantitative methods (in the context of K-theory) to prove the classical Baum—Connes conjecture for a specific class of lacunary hyperbolic groups -along the way we’ll give a very brief description of the Baum—Connes conjecture, define lacunary hyperbolicity properly and also discuss the representation theory of such groups in a broader context I’ll poll the audience to see which parts of this they’d most be interested in hearing during the talk. |
