Kropholler EPSRC - Soluble groups and cohomology

Groups measure symmetry. Soluble groups were born at the outset in group theory, being the embodiment of the mathematics required to understand solving equations in one variable. In its beginnings, group theory was motivated by the theory of numbers. But subsequently the intimate connection between group theory and geometry and topology became recognised. Groups describe symmetries of geometrical objects. This descriptive quality can also be used in reverse. We can take problems in abstract group theory and ask what kinds of geometrical objects can be designed in order to understand those groups. Through the course of this process, group actions on spaces becomes important and the ability to construct spaces out of basic building blocks is needed. Many of the tools are embodied in the branch of mathematics called homological algebra. Therefore we are interested in conditions on groups which are motivated by homological questions. In the case of soluble groups, it has long been known that there are connections with algebraic geometry because of the importance of commutative algebra in the analysis of these groups. Now what do these mysterious words 'homology', 'space', 'action' really mean? Think of an anglepoise lamp. It many different positions are described by means of points in a space sometimes called a phase space, but in pure mathematics it is often called a classifying space. If we have an anglepoise lamp plugged in to the wall, and we wind it around and around, the wire will become wound up around the base: we call that the winding number and we measure using a technique called homology or homotopy. The process of winding the lamp is called an action. So the words can be illustrated with a simple example. But there can also be more complicated examples: the phase space of a system of particles making up a universe. In the theory of soluble groups we often have to be highly innovative in order to find good ways to describe classifying spaces and group actions and that makes the subject especially attractive study. Soluble group theory can only be properly understood in the wider context in which it sits. A great deal is known about the classifying spaces of soluble groups and there is a great deal further to discover. Classifying spaces should be thought of as similar to configuration spaces in theoretical physics: they often play essentially the same role. For example, the configurations of finite sets of integer weighted points on a sphere can be thought of, on the the one hand, as a simplistic model of possible configurations of a universe of particles and anit-particles, and on the other hand as a classifying space for the abelian group representing the homology of the sphere. These connections, when taken seriously, both motivate the subject and inspire new ways of thinking about it and new directions for research. Those innovations are central to this application.