Automorphisms of nilpotent groups and geometric structures

This project, with the title Automorphisms of nilpotent groups and geometric structures has the aim to begin an in-depth study of automorphism groups of (virtually) nilpotent groups arising in many important problems in geometry and topology (e.g., arising in the study of nilmanifolds and infra-nilmanifolds, spectral geometry of Riemannian manifolds and Nielsen fixed-point theory). Very often such problems boil down to the study of specific properties of automorphism groups of nilpotent groups. Usually then some ad hoc methods are applied to solve the particular question. The innovative part of our project lies in the belief that there is a lack of a more general approach, which could be used in several specific situations. Therefore we propose here to start with an in-depth investigation of the automorphism group itself and study several structural aspects of the automorphism group and develop enough computational tools which can then succesfully be applied in a broad range of topological and geometric contexts both by ourselves as by other researchers. The methods to be used are mostly of group-theoretical nature, but also include polynomial functors and eigenvalue techiques. In fact, certain eigenvalue properties of the automorphisms in question yield valuable information for the associated problems in fixed point theory, expanding maps, Anosov diffeomorphisms, and to the study of the algebraic structure of the automorphism group itself.

This project belongs to the area of algebra and geometry in the field of basic scientific research. In particular, it deals with the connection and interaction of these two areas. When talking about geometry one usually thinks of figures in space, like lines, circles, cubes, spheres, curves, surfaces and so on. When a mathematician is working with these objects then he needs to be able to describe them in a very precise way, so that one can understand their properties and interactions. For this it is necessary that one can compute the objects and their interactions. However, it turns out that it is very difficult to do this only with the geometric objects themselves. In fact, it is impossible to ``compute'' with lines, circles, ellipses etc. directly. \\[0.2cm] To resolve this obstacle, one tries to introduce algebraic systems allowing a translation of the geometric problems into computable algebraic problems. The most well-known system here is the coordination system. Each point in the plane can be described by two numbers, given by the horizontal and vertical position. In the same way, each point in three-space can be described by three numbers and so on. This way one can describe lines, circles, ellipses and other things by algebraic equations and is able to do computations. Of course, one can go much further than just considering a coordinate system. \\[0.2cm] In our project we do not use only numbers, but also more complicated structures, called {\em groups and algebras}. They are very well suited to describe problems in geometry and make them computable. We want to study such groups and algebras arising from geometry in detail, in particular Lie groups and Lie algebras, named after the Norwegian mathematician Sophus Lie. Moreover we study ``automorphisms'', which are certain maps preserving a structure. The aim is to obtain fundamental results on problems in geometry by studying the associated algebraic structures.