In the study of the topology or geometry of an infra-nilmanifold, one is often confronted with the question whether or not a certain very specific (auto)morphism of the fundamental group of the manifold exists. These fundamental groups are (virtually) nilpotent groups.
As an example, consider almost inner automorphisms of nilpotent Lie groups, which emerge in the study of isospectral nilmanifolds. An automorphism of a group is said to be almost inner if and only if each group element is conjugate to its image. A nilpotent Lie group which admits a lattice and an almost inner automorphism which is not inner can be used to construct a continuous family of isospectral but non-isometric nilmanifolds. Hence, the existence problem of this kind of automorphisms on a given (virtually) nilpotent group is very interesting. However, up till now, these almost inner automorphisms have not been studied in detail yet.
During my stay at ESI, I want to work further on an algebraic approach towards the study of these almost inner automorphisms. This is done by considering almost inner derivations, the corresponding notion on the Lie algebra level. Only very recently, the first structural results concerning this concept were obtained. However, many things are unknown for the moment.