The Reidemeister spectra of low dimensional crystallographic groups

TL;DR

This paper develops algorithms to compute the Reidemeister spectrum and determine the $R_ fty$-property for automorphisms of crystallographic groups, and applies them to groups up to dimension 6, advancing understanding of their twisted conjugacy classes.

Contribution

It introduces two algorithms for analyzing Reidemeister spectra of crystallographic groups with finite normalisers, and applies these to groups up to dimension 6.

Findings

Algorithms successfully compute Reidemeister spectra.

Determined $R_ Infty$-property for various groups.

Extended analysis to crystallographic groups up to dimension 6.

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 $6$.