The Reidemeister spectrum of low dimensional almost-crystallographic groups

TL;DR

This paper investigates the Reidemeister spectrum of low-dimensional almost-crystallographic groups, identifying those with the $R_ Infty$-property and computing spectra for specific 3- and 4-dimensional groups.

Contribution

It provides a classification of 4-dimensional almost-crystallographic groups with the $R_ Infty$-property and calculates Reidemeister spectra for certain low-dimensional groups.

Findings

Identified which 4D almost-crystallographic groups have the $R_ Infty$-property.

Computed Reidemeister spectra for 3D almost-crystallographic groups.

Calculated spectra for 4D almost-Bieberbach groups.

Abstract

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

Tables

Table 1. Table 1: Conjugacy matrices between representations

Family	$\delta$
1,2
3,4
5
143
146

Table 2. Table 2: MakeList ( 0 , 0 , 0 , 0 0 0 0 0 0,0,0,0 )

$\overline{M}$	$\overline{d}$	$R$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+4$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}+4$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$8\mathbb{N}+8$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$8\mathbb{N}$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$8\mathbb{N}$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$8\mathbb{N}$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+4$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4\mathbb{N}+4$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+4$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}+4$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}$

Table 3. Table 3: MakeList ( 0 , 0 , 0 , 1 0 0 0 1 0,0,0,1 )

$\overline{M}$	$\overline{d}$	$R$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}-2$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4\mathbb{N}-2$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}-2$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}-2$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$8\mathbb{N}$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$8\mathbb{N}$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$8\mathbb{N}$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$8\mathbb{N}$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}-2$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4\mathbb{N}-2$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}-2$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}-2$

Table 4. Table 4: MakeList ( 0 , 0 , 1 , 0 0 0 1 0 0,0,1,0 )

$\overline{M}$	$\overline{d}$	$R$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$8\mathbb{N}+4$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$8\mathbb{N}$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+4$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}$

Table 5. Table 5: MakeList ( 0 , 0 , 1 , 1 0 0 1 1 0,0,1,1 )

$\overline{M}$	$\overline{d}$	$R$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$8\mathbb{N}+4$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$8\mathbb{N}$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}+4$

Table 6. Table 6: MakeList ( 0 , 1 , 1 , 0 0 1 1 0 0,1,1,0 )

$\overline{M}$	$\overline{d}$	$R$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+4$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$8\mathbb{N}+4$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$8\mathbb{N}$

Table 7. Table 7: MakeList ( 0 , 1 , 1 , 1 0 1 1 1 0,1,1,1 )

$\overline{M}$	$\overline{d}$	$R$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}+4$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$8\mathbb{N}+4$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$8\mathbb{N}$

Table 8. Table 8: MakeList ( 1 , 0 , 0 , 0 1 0 0 0 1,0,0,0 )

$\overline{M}$	$\overline{d}$	$R$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4\mathbb{N}-2$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$8\mathbb{N}+6$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}-2$

Table 9. Table 9: MakeList ( 1 , 0 , 0 , 1 1 0 0 1 1,0,0,1 )

$\overline{M}$	$\overline{d}$	$R$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$8\mathbb{N}+2$
	$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}$

Table 10. Table 10: MakeList ( 1 , 0 , 1 , 0 1 0 1 0 1,0,1,0 )

$\overline{M}$	$\overline{d}$	$R$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}-2$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$8\mathbb{N}+6$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}-2$

Table 11. Table 11: MakeList ( 1 , 0 , 1 , 1 1 0 1 1 1,0,1,1 )

$\overline{M}$	$\overline{d}$	$R$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$8\mathbb{N}+2$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4\mathbb{N}$

Table 12. Table 12: MakeList ( 1 , 1 , 1 , 0 1 1 1 0 1,1,1,0 )

$\overline{M}$	$\overline{d}$	$R$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$8\mathbb{N}+2$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}$

Table 13. Table 13: MakeList ( 1 , 1 , 1 , 1 1 1 1 1 1,1,1,1 )

$\overline{M}$	$\overline{d}$	$R$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}-2$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$8\mathbb{N}+6$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}+2$
	$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4\mathbb{N}-2$

Table 14. Table 14: Automorphisms and Reidemeister spectra and for all ( k 1 , k 2 , k 3 , k 4 ) subscript 𝑘 1 subscript 𝑘 2 subscript 𝑘 3 subscript 𝑘 4 (k_{1},k_{2},k_{3},k_{4})

$(k_1,k_2,k_3,k_4)$	$M$	$d$	$R\left(\phi \right)$	${\mathrm{Spec}}_R\left(\Gamma \right)$
$(0,0,0,0)$		$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4m$	$4\mathbb{N}$
$(0,0,0,1)$		$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$2m$	$2\mathbb{N}$
$(0,0,1,0)$		$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4m$	$4\mathbb{N}$
$(0,0,1,1)$		$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4m$	$4\mathbb{N}$
$(0,1,1,0)$		$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$4m$	$4\mathbb{N}$
$(0,1,1,1)$		$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4m$	$4\mathbb{N}$
$(1,0,0,0)$		$\left(\begin{array}{c}0\\ 1\end{array}\right)$	$4m-2$	$4\mathbb{N}-2$
$(1,0,0,1)$		$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$2m+2$	$2\mathbb{N}+2$
$(1,0,1,0)$		$\left(\begin{array}{c}1\\ 0\end{array}\right)$	$4m-2$	$4\mathbb{N}-2$
$(1,0,1,1)$		$\left(\begin{array}{c}1\\ 1\end{array}\right)$	$2m+2$	$2\mathbb{N}+2$
$(1,1,1,0)$		$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$2m+2$	$2\mathbb{N}+2$
$(1,1,1,1)$		$\left(\begin{array}{c}0\\ 0\end{array}\right)$	$4m-2$	$4\mathbb{N}-2$

Equations

$$\forall g,g^{\prime }\in G:g{\sim }_{\phi }{g}^{\prime }⟺\exists h\in G:g=h{g}^{\prime }\phi (h{)}^{-1}.$$

$${\mathrm{Spec}}_R(G):=\{R(\phi )\mid \phi \in \mathrm{Aut}(G)\}.$$

$$\sqrt[G]{H}:=\{g\in G\mid g^k\in H\text{ for some }k\ge 1\}.$$

$$R(\phi )=\infty$$

$$⟺\exists A\in F\text{ such that }\det (\text{mathds}{1}_n-A_{\ast }{D}_{\ast })=0$$

$$⟺\exists A\in F\text{ such that }{A}_{\ast }{D}_{\ast }\text{ has eigenvalue }1.$$

$$R(\phi )=\frac{1}{\#F}\sum _{A\in F}|\det (\text{mathds}{1}_n-A_{\ast }{D}_{\ast })|.$$

$$\left\langle e_{1},e_{2},e_{3},e_{4}\;\Bigg{\rvert}\begin{array}[]{ll}\left[e_{2},e_{1}\right]=1&\left[e_{3},e_{2}\right]=e_{1}^{l_{1}}\\ \left[e_{3},e_{1}\right]=1&\left[e_{4},e_{2}\right]=e_{1}^{l_{2}}\\ \left[e_{4},e_{1}\right]=1&\left[e_{4},e_{3}\right]=e_{1}^{l_{3}}\end{array}\right\rangle.$$

$$\lambda (e_1)$$

$$\lambda (e_3)$$

$$A_{\ast }=\left(\begin{array}{cccc}{ϵ}_A& \ast & \ast & \ast \\ 0& \ast & \ast & \ast \\ 0& \ast & \ast & \ast \\ 0& \ast & \ast & \ast \end{array}\right),D_{\ast }=\left(\begin{array}{cccc}\nu & \ast & \ast & \ast \\ 0& \ast & \ast & \ast \\ 0& \ast & \ast & \ast \\ 0& \ast & \ast & \ast \end{array}\right).$$

$$\text{mathds}{1}_4-A_{\ast }{D}_{\ast }=\left(\begin{array}{cccc}1-\nu {ϵ}_A& \ast & \ast & \ast \\ 0& \ast & \ast & \ast \\ 0& \ast & \ast & \ast \\ 0& \ast & \ast & \ast \end{array}\right).$$

$$\lambda (\Gamma )=\delta \mu ({\Gamma }^{\prime }){\delta }^{-1},$$

$$\left\langle e_{1},e_{2},e_{3},e_{4},\alpha\;\Bigg{\rvert}\begin{array}[]{ll}\left[e_{2},e_{1}\right]=1&\alpha e_{1}=e_{1}\alpha\\ \left[e_{3},e_{1}\right]=1&\alpha e_{2}=e_{1}^{k_{4}}e_{2}^{-1}\alpha\\ \left[e_{4},e_{1}\right]=1&\alpha e_{3}=e_{1}^{k_{5}}e_{3}^{-1}\alpha\\ \left[e_{3},e_{2}\right]=e_{1}^{k_{1}}&\alpha e_{4}=e_{1}^{k_{6}}e_{4}^{-1}\alpha\\ \left[e_{4},e_{2}\right]=e_{1}^{k_{2}}&\alpha^{2}=e_{1}^{k_{7}}\\ \left[e_{4},e_{3}\right]=e_{1}^{k_{3}}&\end{array}\right\rangle,$$

$$\lambda (\alpha )=\left(\begin{array}{ccccc}1& k_4& k_5& k_6& \frac{k_7}{2}\\ 0& -1& 0& 0& 0\\ 0& 0& -1& 0& 0\\ 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 1\end{array}\right).$$

$${\phi }^{\prime }(g⟨e_1⟩)=D^{\prime }(g⟨e_1⟩)=\phi (g)⟨e_1⟩=g^{ϵ}⟨e_1⟩.$$

$$\left\langle e_{1},e_{2},e_{3},e_{4},\alpha\;\Bigg{\rvert}\begin{array}[]{ll}\left[e_{2},e_{1}\right]=1&\alpha e_{1}=e_{1}\alpha\\ \left[e_{3},e_{1}\right]=1&\alpha e_{2}=e_{2}\alpha\\ \left[e_{4},e_{1}\right]=1&\alpha e_{3}=e_{1}^{k_{2}}e_{2}^{-\nu}e_{3}^{-1}\alpha\\ \left[e_{3},e_{2}\right]=1&\alpha e_{4}=e_{1}^{k_{3}}e_{4}^{-1}\alpha\\ \left[e_{4},e_{2}\right]=1&\alpha^{2}=e_{1}^{k_{4}}e_{2}^{\mu}\\ \left[e_{4},e_{3}\right]=e_{1}^{k_{1}}&\end{array}\right\rangle,$$

$$\lambda (\alpha )=\left(\begin{array}{ccccc}1& 0& k_2& k_3& \frac{k_4}{2}\\ 0& 1& -\nu & 0& \frac{\mu }{2}\\ 0& 0& -1& 0& 0\\ 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 1\end{array}\right).$$

$$\phi (e_1)=e_1^{-1},$$

$$\phi (e_2)=e_2^{-1},$$

$$\phi (e_3)=e_1^{k_1-k_2-k_3}{e}_2^{\nu }{e}_3{e}_4^2,$$

$$\phi (e_4)=e_1^{3k_1-k_2-2k_3}{e}_2^{\nu }{e}_3^2{e}_4^3,$$

$$\phi (\alpha )=e_1^{-k_4}{e}_2^{-\mu }\alpha ,$$

$$D_{\ast }=\left(\begin{array}{cccc}-1& \ast & \ast & \ast \\ 0& -1& \ast & \ast \\ 0& 0& 1& 2\\ 0& 0& 2& 3\end{array}\right).$$

$$\left\langle e_{1},e_{2},e_{3},e_{4},\alpha\;\Bigg{\rvert}\begin{array}[]{ll}\left[e_{2},e_{1}\right]=1&\alpha e_{1}=e_{1}\alpha\\ \left[e_{3},e_{1}\right]=1&\alpha e_{2}=e_{2}\alpha\\ \left[e_{4},e_{1}\right]=1&\alpha e_{3}=e_{1}^{k_{2}}e_{4}\alpha\\ \left[e_{3},e_{2}\right]=1&\alpha e_{4}=e_{1}^{k_{3}}e_{2}^{\mu}e_{3}^{-1}e_{4}^{-1}\alpha\\ \left[e_{4},e_{2}\right]=1&\alpha^{3}=e_{1}^{k_{4}}\\ \left[e_{4},e_{3}\right]=e_{1}^{k_{1}}&\end{array}\right\rangle,$$

$$\lambda (\alpha )=\left(\begin{array}{ccccc}1& 0& k_2& -\frac{k_1}{2}+k_3& \frac{k_4}{3}\\ 0& 1& 0& \mu & 0\\ 0& 0& 0& -1& 0\\ 0& 0& 1& -1& 0\\ 0& 0& 0& 0& 1\end{array}\right).$$

$$\left\langle e_{1},e_{2},e_{3}\;\Big{\rvert}\begin{array}[]{ll}\left[e_{2},e_{1}\right]=1&\left[e_{3},e_{2}\right]=e_{1}^{l_{1}}\\ \left[e_{3},e_{1}\right]=1&\end{array}\right\rangle,$$

$$\lambda (e_1)$$

$$\left\langle e_{1},e_{2},e_{3},\alpha\;\Bigg{\rvert}\begin{array}[]{ll}\left[e_{2},e_{1}\right]=1&\alpha e_{1}=e_{1}\alpha\\ \left[e_{3},e_{1}\right]=1&\alpha e_{2}=e_{1}^{k_{2}}e_{2}^{-1}\alpha\\ \left[e_{3},e_{2}\right]=e_{1}^{k_{1}}&\alpha e_{3}=e_{1}^{k_{3}}e_{3}^{-1}\alpha\\ &\alpha^{2}=e_{1}^{k_{4}}\end{array}\right\rangle,$$

$$\lambda (\alpha )=\left(\begin{array}{cccc}1& k_2& k_3& \frac{k_4}{2}\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right).$$

$$\lambda (\phi (\gamma ))=\delta \lambda (\gamma ){\delta }^{-1}.$$

$$\delta =\left(\begin{array}{cccc}-1& n_1& n_2& 0\\ 0& m_1& m_3& d_1/2\\ 0& m_2& m_4& d_2/2\\ 0& 0& 0& 1\end{array}\right),$$

$$\delta \lambda (e_2){\delta }^{-1}$$

$$\delta \lambda (e_3){\delta }^{-1}$$

Full text

This is an Accepted Manuscript of an article published by Taylor & Francis in Experimental Mathematics on 11 Jul 2019, available online: https://www.tandfonline.com/10.1080/10586458.2019.1636426.

1 Introduction

Let $G$ be any group and $\varphi:G\to G$ an endomorphism of this group. Define an equivalence relation $\sim_{\varphi}$ on $G$ given by

[TABLE]

An equivalence class $[g]_{\varphi}$ is called a Reidemeister class of $\varphi$ or $\varphi$-twisted conjugacy class. The Reidemeister number $R(\varphi)$ is the number of Reidemeister classes of $\varphi$ and is therefore always a positive integer or infinity. The Reidemeister spectrum of a group $G$ is the set of all Reidemeister numbers when considering all possible automorphisms of that group:

[TABLE]

If $\operatorname{Spec}_{R}(G)=\{\infty\}$ we say that $G$ has the $R_{\infty}$-property.

Reidemeister numbers originate in Nielsen fixed point theory, where they are defined as the number of fixed point classes of a self-map of a topological space [11], although they also yield applications in algebraic geometry and representation theory [8].

It turns out that many (infinite) groups admit the $R_{\infty}$-property. This is also the case for most almost-crystallographic groups, e.g. in [6] it was shown that $207$ of the $219$ 3-dimensional crystallographic groups and $15$ of the $17$ families of 3-dimensional (non-crystallographic) almost-crystallographic groups all have the $R_{\infty}$-property. Furthermore, in [5] it was shown that $4692$ of the $4783$ 4-dimensional crystallographic groups admit the $R_{\infty}$ property. Moreover, the Reidemeister spectra of all crystallographic groups of dimensions $1$, $2$ and $3$ were calculated, as well as the spectra of the $4$-dimensional Bieberbach groups. In this paper we extend these results by studying the 4-dimensional almost-crystallographic groups.

This paper is structured as follows. In the next two sections, we provide the necessary preliminaries on Reidemeister numbers and almost-crystallographic groups. In section 4 we determine which almost-crystallographic groups of dimension $4$ possess the $R_{\infty}$-property. Sections 5 and 6 are devoted to calculating the Reidemeister spectra of the 3-dimensional almost-crystallographic groups and the 4-dimensional almost-Bieberbach groups respectively. The final section summarises the obtained results.

2 Reidemeister numbers and spectra

In this section we introduce basic notions concerning the Reidemeister number. For a general reference on Reidemeister numbers and their connection to fixed point theory, we refer the reader to [11].

The definitions of the Reidemeister number and Reidemeister spectrum were given in the introduction. However, nothing was said on how we actually determine whether a group has the $R_{\infty}$-property, and if not, how we calculate its Reidemeister spectrum. The following lemma is an essential tool for the former.

Lemma 2.1** (see [8, Section 2.2], [9, Lemma 1.1]).**

Let $N$ be a normal subgroup of a group $G$ and $\varphi\in\operatorname{Aut}(G)$ with $\varphi(N)=N$. We denote the restriction of $\varphi$ to $N$ by $\varphi|_{N}$, and the induced automorphism on the quotient $G/N$ by $\varphi^{\prime}$. We then get the following commutative diagram with exact rows:

{1}$${N}$${G}$${G/N}$${1}$${1}$${N}$${G}$${G/N}$${1}$$\scriptstyle{\varphi|_{N}}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi^{\prime}}

We obtain the following properties:

1. (1)

$R(\varphi)\geq R(\varphi^{\prime})$, 2. (2)

if $R(\varphi^{\prime})<\infty$, $R(\varphi|_{N})=\infty$ and $|\operatorname{Fix}(\varphi^{\prime})|<\infty$, then $R(\varphi)=\infty$.

A direct consequence for characteristic subgroups is the following:

Corollary 2.2**.**

Let $N$ be a characteristic subgroup of $G$. If either

1. (1)

the quotient $G/N$ has the $R_{\infty}$-property, or 2. (2)

$N$* has finite index in $G$ and has the $R_{\infty}$-property,*

then $G$ has the $R_{\infty}$-property as well.

3 Almost-crystallographic groups

Let $G$ be a connected, simply connected, nilpotent Lie group with automorphism group $\operatorname{Aut}(G)$. The affine group $\operatorname{Aff}(G)$ is the semi-direct product $\operatorname{Aff}(G)=G\rtimes\operatorname{Aut}(G)$, where multiplication is defined by $(d_{1},D_{1})(d_{2},D_{2})=(d_{1}D_{1}(d_{2}),D_{1}D_{2})$. If $C$ is a maximal compact subgroup of $\operatorname{Aut}(G)$, then $G\rtimes C$ is a subgroup of $\operatorname{Aff}(G)$. A cocompact discrete subgroup $\Gamma$ of $G\rtimes C$ is called an almost-crystallographic group modelled on the Lie group $G$. The dimension of $\Gamma$ is defined as the dimension of $G$.

If $\Gamma$ is torsion-free, then it is called an almost-Bieberbach group. If $G=\mathbb{R}^{n}$, then it is called a crystallographic group, or a Bieberbach group if it also torsion-free.

Crystallographic groups were historically studied first, and are well understood by the three Bieberbach theorems. These theorems have since been generalised to almost-crystallographic groups, which we will briefly discuss below. We refer to [14] and [3] for more information on the original and generalised theorems respectively.

The generalised first Bieberbach theorem says that if $\Gamma\subseteq\operatorname{Aff}(G)$ is an $n$-dimensional almost-crystallographic group, then its translation subgroup $N:=\Gamma\cap G$ is a uniform lattice of $G$ and is of finite index in $\Gamma$. Moreover, $N$ is the unique maximal nilpotent normal subgroup of $\Gamma$, and is therefore characteristic in $\Gamma$. The quotient group $F:=\Gamma/N$ is a finite group called the holonomy group of $\Gamma$. In fact $F=\{A\in\operatorname{Aut}(G)\;|\;\exists a\in G:\,(a,A)\in\Gamma\}$. If $\Gamma$ is crystallographic ($G=\mathbb{R}^{n}$), we may assume that $N=\mathbb{Z}^{n}$ and $F$ is a subgroup of $\operatorname{GL}_{n}(\mathbb{Z})$.

The generalised second Bieberbach theorem tells us more about automorphisms of almost-crystallographic groups.

Theorem 3.1** (generalised second Bieberbach theorem).**

Let $\varphi:\Gamma\to\Gamma$ be an automorphism of an almost-crystallographic group $\Gamma\subseteq\operatorname{Aff}(G)$ with holonomy group $F$. Then there exists a $(d,D)\in\operatorname{Aff}(G)$ such that $\varphi(\gamma)=(d,D)\circ\gamma\circ(d,D)^{-1}$ for all $\gamma\in\Gamma$. To shorten notation, we will write $\varphi=\xi_{(d,D)}$.

An automorphism $\Phi:G\to G$ of a Lie group $G$ induces an automorphism $\Phi_{*}:{\mathfrak{g}}\to{\mathfrak{g}}$ of the associated Lie algebra ${\mathfrak{g}}$. We will henceforth always denote an induced automorphisms on a Lie algebra with a star ($*$) subscript, for example $A_{*}$ is the Lie algebra automorphism induced by some $A\in F$ where $F\subseteq\operatorname{Aut}(G)$ is the holonomy group of an almost-crystallographic group. In particular, an automorphism $\varphi=\xi_{(d,D)}$ of an almost-crystallographic group has an associated matrix $D_{*}$.

The generalised third Bieberbach theorem is less straightforward to generalise. Unlike for crystallographic groups, it is not true that there are only finitely many $n$-dimensional almost-crystallographic groups for a given dimension $n$. However, we can state that for a given finitely generated torsion-free nilpotent group $N$, there are (up to isomorphism) only finitely many almost-crystallographic groups $\Gamma$ such that the translation subgroup of $\Gamma$ is isomorphic to $N$.

In [3, Section 2.5], this generalisation is proved using the concept of an isolator, which shall prove useful to us as well.

Definition 3.2**.**

Let $G$ be a group with subgroup $H$. The isolator of $H$ in $G$ is defined as

[TABLE]

Although much can be said about isolators, for the purposes of this paper we only care about a very specific result.

Lemma 3.3** (see [3, Lemma 2.4.2]).**

Let $\Gamma$ be an almost-crystallographic group with translation subgroup $N$ of nilpotency class $c$. Then the isolator $\sqrt[N]{\gamma_{c}(N)}\leq Z(N)$ is a characteristic subgroup of $\Gamma$. Moreover, the quotient group $\Gamma/\sqrt[N]{\gamma_{c}(N)}$ is an almost-crystallographic group whose translation subgroup $N/\sqrt[N]{\gamma_{c}(N)}$ has nilpotency class $c-1$. If $c=2$, then this quotient is a crystallographic group.

We will now give the most important results for Reidemeister theory applied to almost-crystallographic groups. A first result allows us to easily determine whether an almost-crystallographic group admits the $R_{\infty}$-property or not.

Theorem 3.4** (see [6, Corollary 3.10]).**

Let $\Gamma$ be an $n$-dimensional almost-crystallographic group with holonomy group $F\subseteq\operatorname{Aut}(G)$ and $\varphi=\xi_{(d,D)}\in\operatorname{Aut}(\Gamma)$ (where we use the notation of theorem 3.1). Then

[TABLE]

The second result only holds for almost-Bieberbach groups, and allows for an easy computation of the Reidemeister number of an automorphism.

Theorem 3.5** (averaging formula, see [10, Theorem 6.11] and [12, Theorem 4.3]).**

Let $\Gamma$ be an $n$-dimensional almost-Bieberbach group with holonomy group $F\subseteq\operatorname{Aut}(G)$, and $\varphi=\xi_{(d,D)}\in\operatorname{Aut}(\Gamma)$ with $R(\varphi)<\infty$. Then

[TABLE]

In general, this formula does not hold for automorphisms of almost-crystallographic groups, examples can be found in [5] and later in this paper. Therefore, the calculation of the Reidemeister spectra usually requires a deeper understanding of how the Reidemeister classes are formed in a specific group.

4 The $R_{\infty}$-property for 4-dimensional almost-crystallographic groups

Every almost-crystallographic group of dimension $1$ or $2$ is crystallographic. In [6] it was determined which $3$-dimensional almost-crystallographic groups admit the $R_{\infty}$-property. We extend these results to dimension $4$. In this case the translation subgroup $N$ is a finitely generated, torsion-free, nilpotent group of rank $4$ and nilpotency class at most $3$. Nilpotency class $1$ is of course the crystallographic case, which was done in [5].

4.1 Nilpotency class $2$

Let $\Gamma$ be an almost-crystallographic group whose translation subgroup $N$ is a nilpotent group of rank $4$ and nilpotency class $2$. In [3] it was shown that $N$ can be given the following presentation:

[TABLE]

Moreover, let $G$ be the Lie group that $\Gamma$ is modelled on. By [2, Theorem 4.1], there exists a faithful affine representation $\lambda:G\rtimes\operatorname{Aut}(G)\to\operatorname{Aff}(\mathbb{R}^{4})$ such that its restriction to $\Gamma$ is again a faithful affine representation. In particular,

[TABLE]

where the values of $l_{1}$, $l_{2}$ and $l_{3}$ are determined by the relations $[e_{3},e_{2}]=e_{1}^{l_{1}}$, $[e_{4},e_{2}]=e_{1}^{l_{2}}$ and $[e_{4},e_{3}]=e_{1}^{l_{3}}$.

Lemma 3.3 tells us that the subgroup $\langle e_{1}\rangle=\sqrt[N]{\gamma_{2}(N)}$ is characteristic and the quotient $\Gamma^{\prime}:=\Gamma/\langle e_{1}\rangle$ is a $3$-dimensional crystallographic group. Using corollary 2.2, we know that if $\Gamma^{\prime}$ has the $R_{\infty}$-property, then so does $\Gamma$. In [3, 4] the almost-crystallographic groups were classified into families based on which crystallographic group $\Gamma^{\prime}$ is. Since only twelve $3$-dimensional crystallographic groups do not have the $R_{\infty}$-property, we need only consider the corresponding twelve families of $4$-dimensional almost-crystallographic groups.

Each of these families can be split in smaller subfamilies, determined by the action of $F$ on $\sqrt[N]{\gamma_{2}(N)}$: every $A\in F$ acts on $e_{1}$ by ${}^{A}e_{1}=e_{1}^{\epsilon_{A}}$ with $\epsilon_{A}\in\{-1,1\}$. The following proposition quickly deals with the subfamilies where $F$ does not act trivially on $\sqrt[N]{\gamma_{2}(N)}$.

Proposition 4.1**.**

Let $\Gamma$ be an almost-crystallographic group with translation subgroup $N$ of rank $4$ and nilpotency class $2$, and holonomy group $F$. If $F$ acts non-trivially on $\sqrt[N]{\gamma_{2}(N)}$, then $\Gamma$ has the $R_{\infty}$-property.

Proof.

Let $A\in F$ arbitrary and $\varphi=\xi_{(d,D)}\in\operatorname{Aut}(\Gamma)$. Since $A$ acts on $\langle e_{1}\rangle=\sqrt[N]{\gamma_{2}(N)}$ by ${}^{A}e_{1}=e_{1}^{\epsilon_{A}}$ with $\epsilon_{A}\in\{-1,1\}$ and $\varphi(e_{1})=e_{1}^{\nu}$ with $\nu\in\{-1,1\}$, $A_{*}$ and $D_{*}$ must have the following forms:

[TABLE]

Thus, $\mathds{1}_{4}-A_{*}D_{*}$ is of the form

[TABLE]

Now let us look at specific $A\in F$. First, let $A$ be the neutral element of $F$, which necessarily acts trivially on $e_{1}$. The above matrix then has upper left entry $1-\nu$, hence $\det(\mathds{1}_{4}-D_{*})\neq 0$ if and only if $\nu=-1$.

Second, let $A$ be an element of $F$ for which $\epsilon_{A}=-1$. Such element exists since we assumed $F$ acts non-trivially on $\sqrt[N]{\gamma_{2}(N)}$. Then the matrix $\mathds{1}_{4}-A_{*}D_{*}$ has upper left entry $1+\nu$, and $\det(\mathds{1}_{4}-A_{*}D_{*})\neq 0$ if and only if $\nu=1$.

As $\nu$ cannot be $-1$ and $1$ at the same time, we always have some $A\in F$ for which $\det(\mathds{1}_{4}-A_{*}D_{*})=0$, and by theorem 3.4 this means that $R(\varphi)=\infty$. Since this holds for any automorphism, $\Gamma$ has the $R_{\infty}$-property. ∎

From the proof of the theorem above, we can also conclude the following:

Proposition 4.2**.**

Let $\Gamma$ be an almost-crystallographic group with translation subgroup $N$ of rank $4$ and nilpotency class $2$, and let $e_{1}$ be a generator of $\sqrt[N]{\gamma_{2}(N)}$. If $\varphi\in\operatorname{Aut}(\Gamma)$ has finite Reidemeister number, then $\varphi(e_{1})=e_{1}^{-1}$.

We will number the twelve families under consideration according to the crystallographic group $\Gamma/\sqrt[N]{\gamma_{2}(N)}$, using the classification in the International Tables in Crystallography [1]: they are families $1$-$5$, $16$, $19$, $22$-$24$, $143$ and $146$. When we write $\Gamma_{n/m}$, we mean the $n$-dimensional crystallographic group with IT-number $m$.

Using the techniques in [3, Section 5.4], we find that for an almost-crystallographic group belonging to one of the families $16$, $19$ or $22$-$24$, $F$ acting trivially on $\sqrt[N]{\gamma_{2}(N)}$ implies that the group is actually crystallographic. Therefore we may omit these families and we are left with only $7$ families to study.

Note that the presentations given in this paper may vary from those in [3, 4]. Let $\Gamma$ and $\lambda$ denote a group and its faithful representation as given in this paper, and let $\Gamma^{\prime}$ and $\mu$ be the corresponding group and representation as given by [3] or [4]. Table 1 contains a matrix $\delta$ such that

[TABLE]

hence $\lambda(\Gamma)$ and $\mu(\Gamma^{\prime})$ are conjugate subgroups of $\operatorname{Aff}(\mathbb{R}^{4})$ and therefore $\Gamma$ and $\Gamma^{\prime}$ are isomorphic.

Family 1.

This family consists of the finitely generated, torsion-free, nilpotent groups of nilpotency class $2$ and rank $4$. It was shown in [7, Section 3.2] that these groups do not have the $R_{\infty}$-property.

Family 2.

Every group in this family has a presentation of the form

[TABLE]

and the faithful representation $\lambda$ is given by

[TABLE]

Set $k:=\gcd(k_{1},k_{2},k_{3})$ and $g:=e_{2}^{k_{3}/k}e_{3}^{-k_{2}/k}e_{4}^{k_{1}/k}$, then the centre $Z(N)$ of the translation subgroup is generated by $e_{1}$ and $g$. Let $\varphi:\Gamma\to\Gamma$ be any automorphism. Since $\langle e_{1}\rangle$ and $Z(N)$ are both characteristic in $\Gamma$, we have that $\varphi(g)=g^{\epsilon}e_{1}^{m}$ for some $\epsilon\in\{-1,1\}$ and $m\in\mathbb{Z}$. Consider the induced automorphism $\varphi^{\prime}=\xi_{(d^{\prime},D^{\prime})}$ on $\Gamma/\langle e_{1}\rangle\cong\Gamma_{3/2}$. Then

[TABLE]

Depending on the value of $\epsilon$, $D^{\prime}_{*}$ has either eigenvalue $1$, in which case $\det(\mathds{1}_{3}-D^{\prime}_{*})=0$, or eigenvalue $-1$, in which case $\det(\mathds{1}_{3}+D^{\prime}_{*})=0$. Since the holonomy group of $\Gamma_{3/2}$ is $\{\mathds{1}_{3},-\mathds{1}_{3}\}$, we obtain by theorem 3.4 that $R(\varphi^{\prime})=\infty$ and by lemma 2.1 that therefore $R(\varphi)=\infty$. Since this holds for an arbitrary automorphism, $\Gamma$ has the $R_{\infty}$-property.

Families 3, 4 and 5.

Every group in one of these families has a presentation of the form

[TABLE]

and the faithful representation $\lambda$ is given by

[TABLE]

Family $3$ is given by $\mu,\nu=0$, family $4$ by $\mu=1,\nu=0$ and family $5$ by $\mu=0,\nu=1$. Define an automorphism $\varphi=\xi_{(d,D)}$ by

[TABLE]

then $D_{*}$ is of the form

[TABLE]

We can apply theorem 3.4 to show that $R(\varphi)<\infty$ and hence $\Gamma$ does not have the $R_{\infty}$-property.

Families 143 and 146.

Every group in one of these families has a presentation of the form

[TABLE]

and the faithful representation $\lambda$ is given by

[TABLE]

Family $143$ is given by $\mu=0$ and family $146$ by $\mu=1$. Using an argument identical to the proof of [6, Theorem 4.4, family 13], we may conclude that all groups in these families have the $R_{\infty}$-property.

4.2 Nilpotency class $3$

By an argument analogous to [9, Example 5.2], a finitely-generated, torsion-free, nilpotent group of nilpotency class $3$ and rank $4$ has the $R_{\infty}$-property. Applying corollary 2.2 then proves that every $4$-dimensional almost-crystallographic group with translation subgroup of nilpotency class $3$ has the $R_{\infty}$-property.

5 The Reidemeister spectra of the 3-dimensional almost-crystallographic groups

Let $\Gamma$ be an almost-crystallographic group whose translation subgroup $N$ is a nilpotent group of rank $3$ and nilpotency class $2$. Such $N$ can be given the following presentation:

[TABLE]

with $l_{1}>0$. Moreover, let $G$ be the Lie group that $\Gamma$ is modelled on. By [2, Theorem 4.1], there exists a faithful affine representation $\lambda:G\rtimes\operatorname{Aut}(G)\to\operatorname{Aff}(\mathbb{R}^{3})$ such that its restriction to $\Gamma$ is again a faithful affine representation. In particular,

[TABLE]

where the value of $l_{1}$ is determined by the relation $[e_{3},e_{2}]=e_{1}^{l_{1}}$. Like in section 4.1, we have that the subgroup $\langle e_{1}\rangle=\sqrt[N]{\gamma_{2}(N)}$ is characteristic in $\Gamma$, and an automorphism $\varphi$ must satisfy $\varphi(e_{1})=e_{1}^{-1}$ to have finite Reidemeister number.

As mentioned before, in [6, Theorem 4.4] it was shown that there are only $2$ families of almost-crystallographic groups that do not admit the $R_{\infty}$-property. We again number these families according to the IT-number of the quotient $\Gamma/\sqrt[N]{\gamma_{2}(N)}$.

Family 1.

The groups in this family are exactly the finitely generated, torsion-free, nilpotent groups of nilpotency class $2$ and rank $3$. In [13, Section 3] it was shown that these groups have Reidemeister spectrum $2\mathbb{N}\cup\{\infty\}$. This was shown specifically for the case $k_{1}=1$, but the argument holds for any $k_{1}>0$.

Family 2.

Every group in this family has a presentation of the form

[TABLE]

and the faithful representation $\lambda$ is given by

[TABLE]

Let $\varphi$ be an automorphism with finite Reidemeister number $R(\varphi)$. Under the representation $\lambda$, this automorphism will correspond to a matrix $\delta\in\operatorname{Aff}(\mathbb{R}^{4})$ such that

[TABLE]

for all $\gamma\in\Gamma$. Since we assumed that $R(\varphi)<\infty$, we have that $\varphi(e_{1})=e_{1}^{-1}$. Moreover, $\varphi$ induces an automorphism $\varphi^{\prime}$ on $\Gamma^{\prime}:=\Gamma/\langle e_{1}\rangle$. Thus, $\delta$ must be of the form

[TABLE]

where the constants $m_{i}$, $d_{j}$ are integers, $m_{1}m_{4}-m_{2}m_{3}=-1$ and $n_{1},n_{2}\in\mathbb{R}$. Using a computer, one can calculate the (unique) values of $n_{1},n_{2}$ and $l_{1},l_{2},l_{3}$ such that

[TABLE]

From the obtained values of $l_{1}$, $l_{2}$ and $l_{3}$, we get

[TABLE]

where all exponents must be integers. This places four conditions on the $m_{i}$ and $d_{j}$:

1. (a)

$k_{1}(m_{1}m_{2}+m_{1}d_{2}-m_{2}d_{1})-k_{2}(m_{1}+1)-k_{3}m_{2}\equiv 0\mod 2$, 2. (b)

$k_{1}(m_{3}m_{4}+m_{3}d_{2}-m_{4}d_{1})-k_{2}m_{3}-k_{3}(m_{4}+1)\equiv 0\mod 2$, 3. (c)

$k_{1}d_{1}d_{2}-k_{2}d_{1}-k_{3}d_{2}\equiv 0\mod 2$, 4. (d)

$m_{1}m_{4}-m_{2}m_{3}=-1$.

For ease of notation, let us set

[TABLE]

We will determine $R(\varphi)$ in a very similar way to the proof of [5, Proposition 5.11]. Let $[x]_{\varphi}$ be a Reidemeister class of $\Gamma$, then for any $k\in\mathbb{Z}$,

[TABLE]

therefore $x\sim_{\varphi}xe_{1}^{2k}$ for all $k\in\mathbb{Z}$. Consider the quotient group $\Gamma^{\prime}=\Gamma/\langle e_{1}\rangle$ and let $\varphi^{\prime}=\xi_{(d/2,M)}$ be the induced automorphism on this quotient. Since we assumed that $R(\varphi)<\infty$, we have that $R(\varphi^{\prime})<\infty$ as well. [5, Proposition 5.10] tells us that $R(\varphi^{\prime})=|\operatorname{tr}(M)|+O(\mathds{1}_{2}-M,d)$ with

[TABLE]

where the bar-notation denotes the element-wise projection to $\mathbb{Z}_{2}$. A Reidemeister class $[x\langle e_{1}\rangle]_{\varphi^{\prime}}$ of $\Gamma^{\prime}$ will lift to at most $2$ Reidemeister classes of $\Gamma$: $[x]_{\varphi}$ and $[xe_{1}]_{\varphi}$; so the number of lifts is either $2$ (when $x\not\sim_{\varphi}xe_{1}$) or $1$ (when $x\sim_{\varphi}xe_{1}$). The latter happens if and only if

[TABLE]

Projecting this to the quotient $\Gamma^{\prime}$, we have

[TABLE]

Since $e_{1}$ is central in $\Gamma$ and $x$ appears exactly once on each side of the equality sign in (1), the $e_{1}$-component of $x$ does not matter. Set $x=e_{2}^{x_{2}}e_{3}^{x_{3}}\alpha^{\epsilon_{x}}$ and $z=e_{1}^{z_{1}}e_{2}^{z_{2}}e_{3}^{z_{3}}\alpha^{\epsilon_{z}}$. Let us first assume that $\epsilon_{z}=0$, then (2) is equivalent to

[TABLE]

with $A$ the holonomy part of $x\langle e_{1}\rangle$. As $R(\varphi^{\prime})<\infty$, we must have $z_{2}=z_{3}=0$. But then $z=e_{1}^{z_{1}}$, and (1) then becomes $xe_{1}=xe_{1}^{2z_{1}}$. As $z_{1}$ is an integer, this is impossible. So, let us assume that $\epsilon_{z}=1$. Writing out (1) component-wise, we find that this condition is equivalent to the following:

1. There exist $z_{1},z_{2},z_{3}\in\mathbb{Z}$ such that:
2. (i)

$2\begin{pmatrix}x_{2}\\ x_{3}\end{pmatrix}=\left(\mathds{1}_{2}-(-1)^{\epsilon_{x}}M\right)\begin{pmatrix}z_{2}\\ z_{3}\end{pmatrix}-(-1)^{\epsilon_{x}}d$, 3. (ii)

$k_{1}z_{2}z_{3}-k_{2}z_{2}-k_{3}z_{3}-k_{4}+1=2z_{1}$.

Condition (i) is independent of the $e_{1}$-components, and hence can be interpreted in terms of the quotient group $\Gamma^{\prime}$. In the proof of [5, Proposition 5.11] it was shown that, for a fixed value of $\epsilon_{x}$, the number of Reidemeister classes $[x\langle e_{1}\rangle]_{\varphi^{\prime}}$ for which a pair $(z_{2},z_{3})$ satisfying (i) exists is exactly $O(\mathds{1}_{2}-M,d)$, i.e. the number of solutions $(\bar{z}_{2},\bar{z}_{3})\in\mathbb{Z}_{2}^{2}$ of the linear system of equations

[TABLE]

Note that the above equation is exactly condition (i) taken modulo $2$.

Since $\epsilon_{x}$ can take two values ($1$ and $-1$), there are in total $2O(\mathds{1}_{2}-M,d)$ Reidemeister classes $[x\langle e_{1}\rangle]_{\varphi^{\prime}}$ satisfying condition (i). On the other hand, there are $|\operatorname{tr}(M)|-O(\mathds{1}_{2}-M,d)$ Reidemeister classes of $\Gamma^{\prime}$ for which condition (i) does not hold (see [5, Section 5]).

Recall that the variable $z_{1}$ appears only in condition (ii). If we have a Reidemeister class $[x\langle e_{1}\rangle]_{\varphi^{\prime}}$ and a pair $(z_{2},z_{3})$ for which (i) holds, then we can find a $z_{1}\in\mathbb{Z}$ to make condition (ii) hold if and only if

[TABLE]

which is exactly condition (ii) taken modulo $2$.

We partition the solutions of (i’) into those that do not satisfy condition (ii’) and those that do. Let $S$ be the number of the former and $T$ the number of the latter, then $S+T=O(\mathds{1}_{2}-M,d)$. Of the $2O(\mathds{1}_{2}-M,d)$ Reidemeister classes $[x\langle e_{1}\rangle]_{\varphi^{\prime}}$ satisfying condition (i), $2S$ lift to two distinct Reidemeister classes $[x]_{\varphi}$ and $[xe_{1}]_{\varphi}$, and $2T$ lift to a single Reidemeister class $[x]_{\varphi}$. All together, we have

[TABLE]

In particular, we get that $R(\varphi)\in 2\mathbb{N}$. Taking the parity of $\operatorname{tr}(M)$ into account, we can further determine the possible Reidemeister numbers:

[TABLE]

where

[TABLE]

There is one special case, however. If $M\equiv\mathds{1}_{2}\mod 2$ all entries of $\mathds{1}_{2}-M$ will be multiples of $2$; so $|\det(\mathds{1}_{2}-M)|=|\operatorname{tr}(M)|\in 4\mathbb{N}$ and therefore $R(\varphi)\in 8\mathbb{N}+2S$.

For a fixed group $\Gamma$ in this family (i.e. a fixed $4$-tuple of parameters $(k_{1},k_{2},k_{3},k_{4})$), an automorphism $\varphi\in\operatorname{Aut}(\Gamma)$ is uniquely determined by the matrix $M\in\operatorname{GL}_{2}(\mathbb{Z})$ and the vector $d\in\mathbb{Z}^{2}$. Our goal is to find out, for each group in the family (or equivalently, for each tuple $(k_{1},k_{2},k_{3},k_{4})$), which $M$ and $d$ satisfy conditions (a) - (d) and thus produce an automorphism.

Conditions (a) - (c) are actually conditions over $\mathbb{Z}_{2}$, and none of the parameters $k_{i}$ appear in condition (d). Therefore, only the parity of the $k_{i}$ will play a role, so we need to check $16$ cases, each corresponding to an element of $\mathbb{Z}_{2}^{4}$. Furthermore, a group with parameters $(k_{1},k_{2},k_{3},k_{4})$ is isomorphic to the group with parameters $(-k_{1},k_{3},k_{2},k_{4})$, which allows us to omit the cases $(0,1,0,0)$, $(0,1,0,1)$, $(1,1,0,0)$ and $(1,1,0,1)$, leaving only $12$ cases. Rather than trying to find all couples $(M,d)$ (of which there are likely to be infinitely many), we can start by finding all couples $(\bar{M},\bar{d})\in\operatorname{GL}_{2}(\mathbb{Z}_{2})\times\mathbb{Z}_{2}^{2}$ satisfying conditions (a)-(c).

The function MakeList defined in algorithm 1 does exactly this. Moreover, it assigns to every couple a set $R$, which is the set of possible Reidemeister numbers the corresponding automorphisms can have. The results can be found in tables 2 to 13. The Reidemeister spectrum of a group is a subset of (or the entirety of) the union of all these sets $R$.

Next, for each quadruplet of parameters, we tried to find a family of automorphisms whose Reidemeister numbers produce the union of these sets $R$. We succeeded in this for every choice of parameters, hence the Reidemeister spectrum always equals the union of the $R$. These automorphisms and their Reidemeister spectra, for all $(k_{1},k_{2},k_{3},k_{4})$, can be found in table 14. For the sake of brevity, we omitted $\infty$ from the spectra in this table.

We may thus conclude that, depending on the parity of the parameters $k_{1}$, $k_{2}$, $k_{3}$ and $k_{4}$, the Reidemeister spectrum is $2\mathbb{N}\cup\{\infty\}$, $4\mathbb{N}\cup\{\infty\}$, $(4\mathbb{N}-2)\cup\{\infty\}$ or $(2\mathbb{N}+2)\cup\{\infty\}$. Note that all almost-Bieberbach groups have parameters with parities $(0,0,0,1)$ and therefore have spectrum $2\mathbb{N}\cup\{\infty\}$.

6 Spectra of 4D almost-Bieberbach groups

We already determined in section 4 which families of four-dimensional almost-crystallographic groups do not have the $R_{\infty}$-property. In [3] it is determined which groups among these families are almost-Bieberbach groups. We use the presentations from section 4.

Family 1.

Every group in this family is a finitely generated, torsion-free, nilpotent group of rank $4$ and nilpotency class $2$. In [7, Section 3.2] it was shown that the Reidemeister spectrum of such group is always $4\mathbb{N}\cup\{\infty\}$.

Family 3.

The almost-Bieberbach groups in this family are those with $(k_{1},k_{2},k_{3},k_{4})=(2k,0,0,1)$ for some $k\in\mathbb{N}$. An automorphism $\varphi=\xi_{(d,D)}$ with $R(\varphi)<\infty$ must be of the form

[TABLE]

with $m_{1}$, $m_{2}$, $m_{3}$, $m_{4}$, $d_{1}$, $d_{2}$, $l$ $\in\mathbb{Z}$ and $m_{1}m_{4}-m_{2}m_{3}=-1$. Then $D_{*}$ is of the form

[TABLE]

Using theorem 3.5, we find that $R(\varphi)=4|m_{1}+m_{4}|\in 4\mathbb{N}$. Now, take the automorphism $\varphi_{m}$ given by

[TABLE]

with $m\in\mathbb{N}$. Then $R(\varphi_{m})=4m$ and hence $\operatorname{Spec}_{R}(\Gamma)=4\mathbb{N}\cup\{\infty\}$.

Family 4.

The almost-Bieberbach groups in this family are those where either $(k_{1},k_{2},k_{3},\allowbreak k_{4})=(k,0,0,0)$ with $k\in\mathbb{N}$ or $(k_{1},k_{2},k_{3},k_{4})=(2k,1,0,0)$ with $k\in\mathbb{N}$. In the former case, such almost-Bieberbach group can be seen as an internal semidirect product $H_{k}\rtimes\mathbb{Z}$, where $H_{k}=\langle e_{1},e_{3},e_{4}\rangle$ and $\mathbb{Z}=\langle\alpha\rangle$. Similarly, in the latter case, a group is an internal semidirect product $H_{2k}\rtimes\mathbb{Z}$.

Both of these semidirect products were studied in [7, Proposition 5.23], their Reidemeister spectra are respectively $4\mathbb{N}\cup\{\infty\}$ and $8\mathbb{N}\cup\{\infty\}$.

Family 5.

The almost-Bieberbach groups in this family are those where $(k_{1},k_{2},k_{3},k_{4})=(k,0,0,1)$ with $k\in\mathbb{N}$. An automorphism $\varphi=\xi_{(d,D)}$ with $R(\varphi)<\infty$ must be of the form

[TABLE]

with $m_{1}$, $m_{2}$, $m_{3}$, $m_{4}$, $d_{1}$, $d_{2}$, $l$ $\in\mathbb{Z}$ and $m_{1}-m_{4}+2m_{1}m_{4}-m_{2}m_{3}=0$. Then $D_{*}$ is of the form

[TABLE]

Using theorem 3.5, we find that $R(\varphi)=8|m_{1}+m_{4}|\in 8\mathbb{N}\cup\{\infty\}$. Now, take the automorphism $\varphi_{m}$ given by

[TABLE]

with $m\in\mathbb{N}$. Then $R(\varphi_{m})=8m$ and hence $\operatorname{Spec}_{R}(\Gamma)=8\mathbb{N}\cup\{\infty\}$.

7 Conclusion

We have determined which (non-crystallographic) almost-crystallographic groups of dimension $4$ admit the $R_{\infty}$ property, and calculated the Reidemeister spectra of the non-crystallographic 3-dimensional almost-crystallographic groups, as well as the spectra of the non-crystallographic 4-dimensional almost-Bieberbach groups. Together with the results of [5], this completes the calculation of the Reidemeister spectra of the $3$-dimensional almost-crystallographic groups and of the $4$-dimensional almost-Bieberbach groups.

Acknowledgement

The author would like to thank the referee for their careful reading and useful suggestions for the paper.