Reidemeister spectra for almost-crystallographic groups

The notion of conjugacy in a group can be generalised to twisted conjugacy. For any endomorphism φ of a group G, we may define an equivalence relation ~ on G by ∀g,g' Є G: g ~ g' if and only if ∃h Є G: g = hg'φ(h)⁻¹. The number of equivalence classes is called the Reidemeister number and is denoted by R(φ). The set of all possible Reidemeister numbers of automorphisms is called the Reidemeister spectrum. This notion originates in topological fixed-point theory. A continuous self-map f on a (sufficiently well-behaved) topological space X induces an endomorphism f* on the fundamental group π₁(X). The Reidemeister number R(f*) is an upper bound for the Nielsen number N(f), which in turn is a lower bound for the number of fixed points of f. In this thesis, we investigate the Reidemeister spectra of almost-crystallographic groups. These groups are generalisations of the crystallographic groups, in the sense that their translation subgroup is nilpotent rather than abelian. The main results can be grouped into two parts. In the first part, we investigate the Reidemeister spectra of finitely generated, torsion-free, nilpotent groups. We compute the spectrum for every such group of dimension at most 4. Furthermore, we compute the Reidemeister spectra of free nilpotent groups of low rank and/or nilpotency class. In the second part, we first determine which low-dimensional almost-crystallographic groups admit automorphisms with finite Reidemeister number. Next, we provide an algorithm that is capable of calculating the Reidemeister number of any given automorphism of a crystallographic group, and use this to calculate the Reidemeister spectra. Finally, we determine which almost-crystallographic groups admit Reidemeister zeta functions, and prove that these functions are rational for groups of dimension at most 3.

Jaar van publicatie:2019

Toegankelijkheid:Open