In this paper we study the number of twisted conjugacy classes (the Reidemeister number) for automorphisms of crystallographic groups. We present two main algorithms for crystallographic groups whose holonomy group has finite normaliser in GLn(Z). The first algorithm calculates whether a group has the R∞-property; the second calculates the Reidemeister spectrum. We apply these algorithms to crystallographic groups up to dimension 6. * Supported by long term structural funding -Methusalem grant of the Flemish Government. Reidemeister numbers and spectrumIn this section we introduce basic notions concerning the Reidemeister number. For a general reference on Reidemeister numbers and its connection with fixed point theory, we refer the reader to [14].

Let M be a nilmanifold with a fundamental group which is free 2-step nilpotent on at least 4 generators. We will show that for any nonnegative integer n there exists a self-diffeomorphism hn of M such that hn has exactly n fixed points and any self-map f of M which is homotopic to hn has at least n fixed points. We will also shed some light on the situation for less generators and also for higher nilpotency classes.

We prove that the Reidemeister zeta functions of automorphisms of crystallographic groups with diagonal holonomy Z2 are rational. As a result, we obtain that Reidemeister zeta functions of automorphisms of almost-crystallographic groups up to dimension 3 are rational. The Reidemeister number and zeta functionIn this section we introduce basic notions concerning Reidemeister numbers. For a general reference on Reidemeister numbers and their connection with fixed point theory, we refer the reader to [8]. In this paper, we use N to denote the set of positive integers and N 0 to denote the set of non-negative integers.Definition 1.1. Let G be a group and ϕ : G → G an endomorphism. Define an equivalence relation ∼ on G by ∀g, g ′ ∈ G : g ∼ g ′ ⇐⇒ ∃h ∈ G : g = hg ′ ϕ(h) −1 .

The Reidemeister number of an endomorphism of a group is the number of twisted conjugacy classes determined by that endomorphism. The collection of all Reidemeister numbers of all automorphisms of a group G is called the Reidemeister spectrum of G. In this paper, we determine the Reidemeister spectra of all fundamental groups of solvmanifolds up to Hirsch length 4.

We determine which non-crystallographic, almost-crystallographic groups of dimension 4 have the R∞-property. We then calculate the Reidemeister spectra of the 3dimensional almost-crystallographic groups and the 4-dimensional almost-Bieberbach groups.