Department of Mathematics
Colloquium
by
Prof. Nigel Higson

It has recently been shown that every countable amenable group admits a "metrically proper" isometric action on an infinite dimensional Euclidean space. Several other classes of groups also so act, including free groups and proper groups of isometries of hyperbolic space. I will try to give a brief account of this topic. I will also outline a proof, due to Gennadi Kasparov and myself, of the so-called Baum-Connes conjecture for groups which act metrically properly on Hilbert space. The conjecture, which belongs to C*-algebra theory, has some important consequences in topology (where the isometry groups in question appear as fundamental groups of manifolds) and at the present time some of these results are accessible only by our C*-algebraic techniques. One example: if a smooth, closed, aspherical manifold has an amenable fundamental group then it cannot admit a metric of positive scalar curvature.