The Reidemeister spectra of low dimensional crystallographic groups

Abstract

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 $\operatorname{GL}_n(\mathbb{Z})$. The first algorithm calculates whether a group has the $R_\infty$-property; the second calculates the Reidemeister spectrum. We apply these algorithms to crystallographic groups up to dimension $6$.