Automorphisms of Nilpotent Groups and Geometric Structures

Up till now many people have encountered a topological or geometrical problem in which they had to answer a question concerning some aspect of the automorphism group of a nilpotent group or Lie algebra. The solutions were then mainly obtained using special techniques, inspired by the specific situation. Instead of also following this ad hoc approach, we propose in this project to start an in-depth study of the automorphism group of nilpotent discrete respectively Lie groups and Lie algebras, keeping in mind that we will be especially interested in those properties of these automorphism groups which are useful in topological or geometrical applications. We will focus on the following three aspects:

1. Combinatorial and Computational aspects
2. Eigenvalues
3. Almost Inner Automorphisms

We will apply the results that we obtained in our study of the automorphisms groups to the following areas:

• Dynamical systems
• self-coverings and co-Hopfian nilpotent groups
• Fixed-point theory
• Affine action of Lie groups and post-Lie algebra structures
• Differential geometry of nilmanifolds