Infinite groups, geometries and operator algebras

Abstract

Examples are of critical importance in operators algebras; the lack of decomposition theorems means we can't use the standard trick of breaking objects down into simple pieces, studying the pieces and inferring information about the whole. Infinite groups acting on geometries enable us to generate a surprisingly large class of examples of operator algebras. While the impact of the work is in the field of operator algebras, I will focus on the groups and the geometries, which is where the action is (pun not originally intended but irresistible once noted).