How does the structure of a group determine its Reidemeister spectrum?

The concept of conjugacy in a group G has a natural generalisation to so-called twisted conjugacy: given an endomorphism φ ∈ End(G), we put an equivalence relation on G by stating that, for all x, y ∈ G, x∼ y if and only if ∃z∈G:xφ(z)=zy. The number of equivalence classes of ∼φ is called the Reidemeister number of φ and the set of all possible Reidemeister numbers is called the Reidemeister spectrum of G, denoted by SpecR(G). If SpecR(G) = {∞}, G is said to have the R∞-property.
Twisted conjugacy arises naturally in Nielsen fixed-point theory, but there is also a strong algebraic interest in twisted conjugacy, in particular in the R∞ -property.
In this thesis, we investigate how structural properties of a group translate into information regarding the Reidemeister spectrum. By ‘structural properties’, we mean properties such as nilpotency, (residual) finiteness, splitting as a direct product and so on.
In the first part of this thesis, we study the behaviour of twisted conjugacy in group constructions. After recalling the known results about twisted conjugacy in general extensions, we focus on central extensions and derive an addition and product formula for Reidemeister numbers. Subsequently, we discuss the Reidemeister spectrum of free products, and we give a first family of groups for which the Reidemeister spectrum of a direct product of such groups is completely determined by the spectra of the individual factors.
The second part deals with nilpotent groups. We use the well-known product formula for Reidemeister numbers on finitely generated torsion-free nilpotent groups to prove relations among Reidemeister numbers of endomorphisms on both the group itself and its finite index subgroups. In addition, we revisit direct products and determine sufficient conditions for the Reidemeister spectrum of a direct product of nilpotent groups to be completely determined by the spectra of the individual factors. Finally, we prove an addition formula for finitely generated nilpotent groups with torsion and use it to calculate the Reidemeister spectra of several families of nilpotent groups.
The third and final part consists of results about finite groups. We first prove two alternative methods to compute the Reidemeister number of an endomorphism on a finite group. Afterwards, we completely compute the Reidemeister spectrum of two families of finite groups: the finite abelian groups and split metacyclic groups of the form Cn ⋊ Cp with p a prime number.