The Reidemeister spectrum and the R-infinity property for residually nilpotent groups

Any automorphism of a group G determines an equivalence relation of twisted conjugacy on G. The equivalence classes are called Reidemeister classes and their number is the Reidemeister number of the automorphism. These notions originate from topological fixed point theory, where they correspond to fixed point classes, but nowadays these appear also in many other branches of mathematics, such as algebraic geometry, cryptography, ... . The Reidemeister spectrum of a group is the collection of all the possible Reidemeister numbers for that group and a group is said to have the R-infinity property if for every automorphism of G, the Reidemeister number is infinite. In the last few decades, there has been a lot of research in computing the Reidemeister spectrum of a group or in proving that a certain family of groups has the R-infinity property. Unfortunately, explicitly determining the complete Reidemeister spectrum of a group turns out to be a very delicate and difficult task. Via this project we therefore propose a new approach that will allow us to study the Reidemeister spectrum from a more qualitative point of view, rather than aiming for a complete description in full details. Concerning the R-infinity property, all results up till now have been obtained for certain families of groups, using ad hoc methods adapted to a specific family. By approaching a given group via nilpotent groups, we propose in this project a first systematic way of studying the R-infinity property.