K-Theory, Group C*-Algebras, Large Scale Geometry, and Topology

Dear Joe, John and I worked up the following paragraphs summarizing our grant proposal. Let me know if you find them unsatisfactory. Yours, Nigel ------------------------------------------------------------------ The large scale geometry of groups and spaces plays a determining role in the calculation of invariants in C*-algebra theory and topology. The investigators aim to explore the effect of the geometry and topology of group boundaries (defined using large scale geometry) on the harmonic analysis of groups and the determination of their C*-algebra K-theory. The Baum-Connes conjecture proposes a means of calculating the K-theory of reduced group C*-algebras which blends group homology with the representation theory of finite subgroups. The conjecture, if true, would have a number of implications in geometry and topology, and a fascinating circle of ideas is coming into view which links the Baum-Connes conjecture to aspects of the harmonic analysis of groups and the geometry of group actions on boundary spaces. The investigators will attempt to clarify these relations. A long term goal is to prove the Baum-Connes conjecture, and more importantly to understand better its meaning, for classes such as the hyperbolic groups of Gromov. More immediate objectives include clarifying the relationships between existing proofs of partial forms of the conjecture for these groups, and developing further the connections between C*-algebra K-theory, manifold theory, and controlled topology. Although the tools used to investigate it are rather elaborate, the idea behind large scale geometry is very simple: ignore the local, small scale fluctuations in a quantity and concentrate on its large scale, or long term, behaviour. By doing so, trends or qualities may become apparent which are obscured by inconsequential, small scale fluctations. The investigators have developed tools to distinguish between different sorts of multi-dimensi onal, large scale behaviour in geometry. Somewhat surprisingly, aside from their intrinsic interest, their tools have found application in ordinary, small scale geometry.