Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups

TL;DR

This paper constructs Bieberbach groups with prescribed finite abelian holonomy from Artin braid groups, providing explicit holonomy representations and exploring geometric structures like Anosov diffeomorphisms and Kähler geometry.

Contribution

It introduces a method to realize Bieberbach groups with specific abelian holonomy from braid groups and describes their holonomy representations explicitly.

Findings

Existence of Bieberbach subgroups with given abelian holonomy

Explicit description of holonomy representations

Conditions for Kähler and Anosov structures on associated flat manifolds

Abstract

Let $n\geq 3$. In this paper we show that for any finite abelian subgroup $G$ of $S_n$ the crystallographic group $B_n/[P_n,P_n]$ has Bieberbach subgroups $\Gamma_{G}$ with holonomy group $G$. Using this approach we obtain an explicit description of the holonomy representation of the Bieberbach group $\Gamma_{G}$. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of $\Gamma_{G}$ and determine the existence of Anosov diffeomorphisms and K\"ahler geometry of the flat manifold ${\cal X}_{\Gamma_{G}}$ with fundamental group the Bieberbach group $\Gamma_{G}\leq B_n/[P_n,P_n]$.

Equations

$$1⟶P_n⟶B_n\stackrel{\sigma }{⟶}{S}_n⟶1.$$

$$1⟶P_n/[P_n,P_n]⟶B_n/[P_n,P_n]\stackrel{\overline{\sigma }}{⟶}{S}_n⟶1,$$

$${\tilde{H}}_n=\frac{{\sigma }^{-1}(H)}{[P_n,P_n]}.$$

$$G_1$$

$$G_2$$

$$X_1$$

$$X_2$$

$$X_3$$

$$X_4$$

$${\delta }_{n_l,p_{l+1}^{r_{l+1}}}={\sigma }_{n_l+p_{l+1}^{r_{l+1}}-1}\cdots {\sigma }_{n_l+\frac{p_{l+1}^{r_{l+1}}+1}{2}}{\sigma }_{n_l+\frac{p_{l+1}^{r_{l+1}}-1}{2}}^{-1}\cdots {\sigma }_{n_l+1}^{-1},$$

$${\eta }_f=\overline{\iota }(1,\dots ,1,{\sigma }_1\cdots {\sigma }_{2^{r_{t+f}}-1},1,\dots ,1)={\sigma }_{n_t+m_{f-1}+1}\cdots {\sigma }_{n_t+m_f-1}$$

$$\overline{\iota }:\frac{B_{p_1^{r_1}}}{[P_{p_1^{r_1}},P_{p_1^{r_1}}]}\times \cdots \times \frac{B_{p_t^{r_t}}}{[P_{p_t^{r_t}},P_{p_t^{r_t}}]}\times \frac{B_{2^{r_{t+1}}}}{[P_{2^{r_{t+1}}},P_{2^{r_{t+1}}}]}\times \cdots \times \frac{B_{2^{r_{t+v}}}}{[P_{r_{t+v}},P_{r_{t+v}}]}⟶\frac{B_n}{[P_n,P_n]}$$

$${\Lambda }_{\text{[odd prime]}}(s_1,\dots ,s_t)$$

$${\Lambda }_{\text{[prime 2]}}(s_{t+1},\dots ,s_{t+v})$$

$$M$$

$$y_{j,k}=M_k{x}_j{\overline{M_k{x}_j}}^{-1}$$

$$M_k=\prod _{\begin{array}{c}0\le s_{\alpha }\le p_{\alpha }^{r_{\alpha }}-1\\ 2\le \alpha \le t\end{array}}{\Lambda }_{\text{odd prime}}(n_1-1,s_2,\dots ,s_t)\prod _{\begin{array}{c}0\le s_{\alpha }\le 2^{r_{\alpha }}-1\\ t+1\le \alpha \le t+v\end{array}}{\Lambda }_{\text{prime 2}}(s_{t+1},\dots ,s_{t+v})$$

$$M_k=\prod _{\begin{array}{c}0\le s_{\alpha }\le p_{\alpha }^{r_{\alpha }}-1\\ 1\le \alpha \le t\end{array}}{\Lambda }_{\text{odd prime}}(s_1,s_2,\dots ,s_t)\prod _{\begin{array}{c}0\le s_{\alpha }\le 2^{r_{\alpha }}-1\\ t+1\le \alpha \le t+v\\ \alpha \ne t+f\end{array}}{\Lambda }_{\text{prime 2}}(s_{t+1},\dots ,2^{r_{t+f}}-1,\dots ,s_{t+v})$$

$$C_1$$

$$I$$

$$C_2=\left\{A_{r,s}^d\in X_2\mid (r,s)\in I\right\}.$$

$$C_1\cup C_2\cup C_3$$

$$w=\theta (A_{1,2}{\delta }_{0,n_1}{)}^{x_1}(A_{n_1+1,n_1+2}{\delta }_{n_1,p_2^{r_2}}{)}^{x_2}\cdots (A_{n_{t-1}+1,n_{t-1}+2}{\delta }_{n_{t-1},p_t^{r_t}}{)}^{x_t}.$$

$$(A_{1,2}{\delta }_{0,p_1^{r_1}}{)}^{p_1^{r_1}}$$

$$=A_{1,2}{A}_{1,p_1^{r_1}}{A}_{p_1^{r_1}-1,p_1^{r_1}}\cdots A_{2,3}.$$

$$A^d=A_{1,2}^{\left(\frac{d}{p_1^{r_1}}\right)x_1}{A}_{1,p_1^{r_1}}^{\left(\frac{d}{p_1^{r_1}}\right)x_1}{A}_{p_1^{r_1}-1,p_1^{r_1}}^{\left(\frac{d}{p_1^{r_1}}\right)x_1}\cdots A_{2,3}^{\left(\frac{d}{p_1^{r_1}}\right)x_1}\cdot \prod _{l=1}^{t-1}(A_{n_l+1,n_l+2}{\delta }_{n_l,p_{l+1}^{r_{l+1}}}{)}^{d{x}_{l+1}}$$

$$\theta =(A_{1,2}\cdots A_{2,3}{)}^{{\lambda }_{1,2}}\cdot \prod _{l=1}^{t-1}{\left((A_{n_l+1,n_l+2}{\delta }_{n_l,p_{l+1}^{r_{l+1}}}{)}^{p_{l+1}^{r_{l+1}}}\right)}^{{\lambda }_{n_l+1,n_l+2}}\cdot \prod _{(r,s)\in I}{A}_{r,s}^{d{\lambda }_{r,s}}\cdot \prod _{A_{i,j}\in A_3}{A}_{i,j}^{{\lambda }_{i,j}}$$

$$1=w^d=(\theta A{)}^d=\theta (A\theta A^{-1})(A^2\theta A^{-2})\cdots (A^{d-1}\theta A^{-(d-1)})A^d.$$

$${\lambda }_{1,2}+({\lambda }_{1,2}+d{\lambda }_{2,3})+\cdots +({\lambda }_{1,2}+d{\lambda }_{1,p_1^{r_1}})+\frac{d}{p_1^{r_1}}{x}_1=0,$$

$${\psi }_q:{\mathbb{Z}}_q⟶\mathrm{Aut}\left({\mathbb{Z}}^{\frac{n(n-1)}{2}}\right)$$

$$A_1$$

$$A_2$$

$$I$$

$$\delta =\prod _{0\le l\le t-1}{\delta }_{n_l,p_{l+1}^{r_{l+1}}}.$$

Full text

Let $n\geq 3$. In this paper we show that for any finite abelian subgroup $G$ of $S_{{n}}$ the crystallographic group $B_{n}/[P_{n},P_{n}]$ has Bieberbach subgroups $\Gamma_{G}$ with holonomy group $G$. Using this approach we obtain an explicit description of the holonomy representation of the Bieberbach group $\Gamma_{G}$. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of $\Gamma_{G}$ and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold ${\cal X}_{\Gamma_{G}}$ with fundamental group the Bieberbach group $\Gamma_{G}\leq B_{n}/[P_{n},P_{n}]$. *

††footnotetext: 2010 Mathematics Subject Classification. Primary: 20F36, 20H15; Secondary: 57N16. Key Words and Phrases. Braid group, Bieberbach group, flat manifold, Anosov diffeomorphism, Kähler manifold

1 Introduction

A surprising connection was given in [6] between crystallographic groups with quotients of the Artin braid group $B_{n}$ by the commutator subgroup of the pure Artin braid group $P_{n}$, $B_{n}/[P_{n},P_{n}]$. That results were extended to the generalised braid group associated to an arbitrary complex reflection group and also was studied the existence of Kähler structures on flat manifolds arising from subquotients of these groups, see [11]. Was proved in [6, Proposition 1] that, for $n\geq 3$, $B_{n}/[P_{n},P_{n}]$ is a crystallographic group of dimension $n(n-1)/2$, with holonomy group the symmetric group $S_{n}$, such that the torsion of the quotient $B_{n}/[P_{n},P_{n}]$ is equal to the odd torsion of $S_{{n}}$ (see [6, Corollary 4]). As a consequence $\widetilde{H}=\frac{\sigma^{-1}(H)}{[P_{n},\,P_{n}]}$ is a Bieberbach subgroup of $B_{n}/[P_{n},P_{n}]$ with holonomy group $H$ if and only if $H$ is a 2-subgroup of $S_{n}$, where $\sigma\colon B_{n}\rightarrow S_{n}$ is the natural projection, see [6, Corollary 13 and Theorem 20]. In [13] was studied Bieberbach groups of the form $\widetilde{H}$ arising from Artin braid groups with cyclic holonomy group $H=\mathbb{Z}_{2^{d}}$, the authors computed the center of the Bieberbach group $\widetilde{H}$, decomposed its holonomy representation in irreps, and with this information the authors were able to determine whether the related flat manifold admits Anosov diffeomorphism and/or Kähler geometry (the last one for even dimension).

In this paper, we show that the quotient group $B_{n}/[P_{n},P_{n}]$ has Bieberbach subgroups of dimension $n(n-1)/2$ with holonomy a finite abelian subgroup of $S_{{n}}$. Our initial motivation emanates from the fact that is not possible to construct Bieberbach subgroups of $B_{n}/[P_{n},P_{n}]$ just taking $\widetilde{H}=\frac{\sigma^{-1}(H)}{[P_{n},\,P_{n}]}$ if $H$ has odd order elements, because the torsion of the quotient $B_{n}/[P_{n},P_{n}]$ is equal to the odd torsion of $S_{{n}}$ (see [6, Corollary 4]). However, in [6, Remarks 26(c)] the authors showed a Bieberbach subgroup $L$ of $B_{3}/[P_{3},P_{3}]$ with holonomy group $\mathbb{Z}_{3}$. We shall follow the idea used in [6, Remarks 26(c)] to exhibit Bieberbach subgroups of $B_{n}/[P_{n},P_{n}]$ with finite abelian holonomy $G\leq S_{{n}}$.

This paper is organised as follows. In Section 2 we recall some definitions and facts about crystallographic and Bieberbach groups, Artin braid groups and the quotients related with crystallographic groups. In Section 3 we will prove the following theorem that guarantees the existence and realise Bieberbach subgroups of $B_{n}/[P_{n},P_{n}]$ with abelian holonomy group contained in $S_{{n}}$.

Theorem 1.

Let $G$ be a finite abelian group.

1. (a)

There exists $n$ and a Bieberbach subgroup $\Gamma_{G}$ of $B_{n}/[P_{n},P_{n}]$ of dimension $n(n-1)/2$ with holonomy group $G$. 2. (b)

The finite abelian group $G$ is the holonomy group of a flat manifold $\mathcal{X}_{\Gamma_{G}}$ of dimension $n(n-\leavevmode\nobreak\ 1)/2$, where $n$ is an integer for which $G$ embeds in the symmetric group $S_{{n}}$, and the fundamental group of $\mathcal{X}_{\Gamma_{G}}$ is isomorphic to a subgroup $\Gamma_{G}$ of $B_{n}/[P_{n},P_{n}]$.

L. Auslander and M. Kuranishi proved that if $G$ is any finite group and $G=F/R$, with $F$ and $R$ free non-abelian, then $F_{0}=F/[R,R]$ is the fundamental group of a flat manifold (see [2, Theorem 3]). As a consequence, they proved that any finite group $G$ is the holonomy group of a flat manifold, see Theorem 6. We note that Theorem 1 reproves Auslander-Kuranishi theorem for finite abelian groups using braid theory.

As an application, motivated by [13], in Section 4 we shall consider the case of cyclic holonomy groups of odd order, i.e. Bieberbach subgroups $\Gamma_{\mathbb{Z}_{q}}$ ($\Gamma_{q}$ for short) with holonomy group $\mathbb{Z}_{q}\leq S_{{n}}$ of odd order. In Subsection 4.1 we describe the holonomy representation of $\Gamma_{q}$ as a matrix representation (see Theorem 11) and using it we prove the following.

Theorem 2.

Let $q=p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots p_{t}^{r_{t}}$ be an odd number, where $p_{i}^{r_{i}}$ are distinct odd primes and $r_{i}\geq 1$ for all $1\leq i\leq t$. Let $\mathcal{X}_{\Gamma_{q}}$ be the flat manifold of dimension $\frac{n(n-1)}{2}$ with fundamental group $\Gamma_{q}\leq B_{n}/[P_{n},P_{n}]$ and holonomy group $\mathbb{Z}_{q}$. Then

1. (a)

$\mathcal{X}_{\Gamma_{q}}$* is orientable.* 2. (b)

The first Betti number of $\mathcal{X}_{\Gamma_{q}}$ is $\beta_{1}(\mathcal{X}_{\Gamma_{q}})=\displaystyle\sum_{i=1}^{t}\frac{p_{i}^{r_{i}}-1}{2}+\frac{t(t-1)}{2}$. 3. (c)

The flat manifold $\mathcal{X}_{\Gamma_{q}}$ with fundamental group $\Gamma_{q}$ admits Anosov diffeomorphism if and only if $q\neq 3$.

A long standing problem in dynamical systems is the classification of all compact manifolds supporting an Anosov diffeomorphism [15]. We note that the proof of Theorem 2 mostly depends on the holonomy representation of the Bieberbach group $\Gamma_{q}$, exploring the eigenvalues of the matrix representation (see equation (20) in Theorem 11). At the end of Subsection 4.1 we give a presentation of the Bieberbach group $\Gamma_{q}$ of Theorem 2 (see Proposition 12) and we also give a set of generators of its center in Theorem 13. Finally, in Subsection 4.2 we obtain the following result.

Theorem 3.

Let $p$ be an odd prime and let $r\geq 1$. Let $\mathcal{X}_{\Gamma_{p^{r}}}$ be the flat manifold of dimension $\frac{p^{r}(p^{r}-1)}{2}$ with fundamental group $\Gamma_{p^{r}}\leq B_{p^{r}}/[P_{p^{r}},P_{p^{r}}]$ and holonomy group $\mathbb{Z}_{p^{r}}$. Then the flat manifold $\mathcal{X}_{\Gamma_{p^{r}}}$ is Kähler if and only if there is an integer $u\geq 1$ such that $p^{r}=4u+1$.

As a consequence if $p$ is an odd prime such that $p^{r}=4u+1$ for some $u,r\geq 1$ then the flat manifold $\mathcal{X}_{\Gamma_{p^{r}}}$ of Theorem 3 is a Calabi-Yau flat manifold of dimension $2u(4u+1)$, see Corollary 15.

2 Preliminaries

In this section, we recall briefly the definition of crystallographic and Bieberbach groups, for more details, see [3, Section I.1.1], [4, Section 2.1] or [17, Chapter 3]. We also recall some facts about the Artin braid group $B_{n}$ on $n$ strings as well as the quotient group $B_{n}/[P_{n},P_{n}]$ studied in [6]. We refer the reader to [7] or [12] for more details about Artin braid groups. From now on, we identify $\operatorname{Aut}(\mathbb{Z}^{n})$ with $\operatorname{GL}(n,\mathbb{Z})$.

2.1 Crystallographic and Bieberbach groups

Let $G$ be a Hausdorff topological group. A subgroup $H$ of $G$ is said to be a discrete subgroup if it is a discrete subset. If $H$ is a closed subgroup of $G$ then the coset space $G/H$ has the quotient topology for the projection $\pi\colon\thinspace G\longrightarrow G/H$, and $H$ is said to be uniform if $G/H$ is compact.

Definition.

A discrete and uniform subgroup $\Pi$ of $\mathbb{R}^{n}\rtimes\operatorname{O}(n,\mathbb{R})\subseteq\operatorname{Aff}(\mathbb{R}^{n})$ is said to be a crystallographic group of dimension $n$. If in addition $\Pi$ is torsion free then $\Pi$ is called a Bieberbach group of dimension $n$.

An integral representation of rank $n$ of $\Phi$ is a homomorphism $\Theta\colon\thinspace\Phi\longrightarrow\operatorname{Aut}(\mathbb{Z}^{n})$. We say that $\Theta$ is a faithful representation if it is injective. The following characterisation of crystallographic groups was used in [6] to stablish the connection between braid groups and crystallographic groups.

Lemma 4 (Lemma 8 of [6]).

*Let $\Pi$ be a group. Then $\Pi$ is a crystallographic group if and only if there exist an integer $n\in\mathbb{N}$ and a short exact sequence

\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\textstyle{\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}

such that:*

1. (a)

$\Phi$* is finite, and* 2. (b)

the integral representation $\Theta\colon\thinspace\Phi\longrightarrow\operatorname{Aut}(\mathbb{Z}^{n})$, induced by conjugation on $\mathbb{Z}^{n}$ and defined by $\Theta(\varphi)(x)=\pi x\pi^{-1}$, where $x\in\mathbb{Z}^{n}$, $\varphi\in\Phi$ and $\pi\in\Pi$ is such that $\zeta(\pi)=\varphi$, is faithful.

If $\Pi$ is a crystallographic group, the integer $n$ that appears in the statement of Lemma 4 is called the dimension of $\Pi$, the finite group $\Phi$ is called the holonomy group of $\Pi$, and the integral representation $\Theta\colon\thinspace\Phi\longrightarrow\operatorname{Aut}(\mathbb{Z}^{n})$ is called the holonomy representation of $\Pi$.

Was stated in [6, Corollary 10] that if $\Pi$ is a crystallographic group of dimension $n$ and holonomy group $\Phi$, and $H$ is a subgroup of $\Phi$, then there exists a crystallographic subgroup of $\Pi$ of dimension $n$ with holonomy group $H$. This statement was used in [6, Corollary 13] to provide other crystallographic subgroups of $B_{n}/[P_{n},P_{n}]$.

A Riemannian manifold $M$ is called flat if it has zero curvature at every point. As a consequence of the first Bieberbach Theorem, there is a correspondence between Bieberbach groups and fundamental groups of closed flat Riemannian manifolds (see [4, Theorem 2.1.1] and the paragraph that follows it).

Remark 5.

We recall that the flat manifold determined by a Bieberbach group $\Pi$ is orientable if and only if the integral representation $\Theta\colon\thinspace\Phi\longrightarrow\operatorname{GL}(n,\mathbb{Z})$ satisfies $\operatorname{\text{Im}}\left({\Theta}\right)\subseteq\operatorname{SO}(n,\mathbb{Z})$. This being the case, we say that $\Pi$ is an orientable Bieberbach group.

It is a natural problem to classify the finite groups that are the holonomy group of a flat manifold. The answer was given by L. Auslander and M. Kuranishi in 1957, see [2, Theorem 3], [17, Theorem 3.4.8] or [3, Theorem III.5.2].

Theorem 6 (Auslander and Kuranishi).

Any finite group is the holonomy group of some flat manifold.

2.2 Artin braid groups

We recall in this subsection some facts about the Artin braid group $B_{n}$ on $n$ strings. We refer the reader to [7] and [12] for more details. It is well known that $B_{n}$ possesses a presentation with generators $\sigma_{1},\ldots,\sigma_{n-1}$ that are subject to the relations $\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}$ for all $1\leq i<j\leq n-1$ such that $\left\lvert i-j\right\rvert\geq 2$ and $\sigma_{i+1}\sigma_{i}\sigma_{i+1}=\sigma_{i}\sigma_{i+1}\sigma_{i}$ for all $1\leq i\leq n-2$.

Let $\sigma\colon\thinspace B_{n}\longrightarrow S_{{n}}$ be the homomorphism defined on the given generators of $B_{n}$ by $\sigma(\sigma_{i})=(i,\,i+1)$ for all $1\leq i\leq n-1$. Just as for braids, we read permutations from left to right so that if $\alpha,\beta\in S_{{n}}$ then their product is defined by $\alpha\cdot\beta(i)=\beta(\alpha(i))$ for $i=1,2,\ldots,n$. The pure braid group $P_{n}$ on $n$ strings is defined to be the kernel of $\sigma$, from which we obtain the following short exact sequence:

[TABLE]

A generating set of $P_{n}$ is given by $\left\{A_{i,j}\right\}_{1\leq i<j\leq n}$, where $A_{i,j}=\sigma_{j-1}\cdots\sigma_{i+1}\sigma_{i}^{2}\sigma_{i+1}^{-1}\cdots\sigma_{j-1}^{-1}$. For $1\leq i<j\leq n$, we also set $A_{j,i}=A_{i,j}$, and if $A_{i,j}$ appears with exponent $m_{i,j}\in\mathbb{Z}$, then we let $m_{j,i}=m_{i,j}$. It follows from the presentation of $P_{n}$ given in [7] that $P_{n}/[P_{n},P_{n}]$ is isomorphic to $\mathbb{Z}^{n(n-1)/2}$, and that a basis is given by the $A_{i,j}$, where $1\leq i<j\leq n$, and where by abuse of notation, the $[P_{n},P_{n}]$-coset of $A_{i,j}$ will also be denoted by $A_{i,j}$. Using equation (1), we obtain the following short exact sequence:

[TABLE]

where $\overline{\sigma}\colon\thinspace B_{n}/[P_{n},P_{n}]\longrightarrow S_{{n}}$ is the homomorphism induced by $\sigma$.

In [6, Theorem 3] it is assumed that $n_{1},\ldots,n_{t}$ are odd integers. Nevertheless the proof of [6, Theorem 3(a)], which states that the inclusion of a direct product $\iota\colon\thinspace B_{n_{1}}\times\cdots\times B_{n_{t}}\longrightarrow B_{m}$ induces an injective homomorphism $\overline{\iota}\colon\thinspace\displaystyle\frac{B_{n_{1}}}{[P_{n_{1}},P_{n_{1}}]}\times\cdots\times\frac{B_{n_{t}}}{[P_{n_{t}},P_{n_{t}}]}\longrightarrow\frac{B_{m}}{[P_{m},P_{m}]}$, do not use the hypotheses that $n_{1},n_{2},\ldots,n_{t}$ are odd integers. So we can state that result as follows.

Theorem 7 (Theorem 3(a) of [6]).

Let $t,m\in\mathbb{N}$, and let $n_{1},n_{2},\ldots,n_{t}$ be integers greater than or equal to $3$ for which $\sum_{i=1}^{t}\,n_{i}\leq m$. Then the inclusion $\iota\colon\thinspace B_{n_{1}}\times\cdots\times B_{n_{t}}\longrightarrow B_{m}$ induces an injective homomorphism $\overline{\iota}\colon\thinspace\displaystyle\frac{B_{n_{1}}}{[P_{n_{1}},P_{n_{1}}]}\times\cdots\times\frac{B_{n_{t}}}{[P_{n_{t}},P_{n_{t}}]}\longrightarrow\frac{B_{m}}{[P_{m},P_{m}]}$.

Theorem 7 will be useful to prove Proposition 10. Since $B_{1}$ is the trivial group and $B_{2}/[P_{2},P_{2}]\cong\mathbb{Z}$, we shall suppose in most of this paper that $n\geq 3$. The study of the action by conjugacy of $B_{n}$ on $P_{n}$ provides the following result.

Proposition 8 (Proposition 12 of [6]).

Let $\alpha\in B_{n}/[P_{n},P_{n}]$, and let $\pi$ be the permutation induced by $\alpha^{-1}$, then $\alpha A_{i,j}\alpha^{-1}=A_{\pi(i),\pi(j)}$ in $P_{n}/[P_{n},P_{n}]$.

Proposition 8 was important to establish the connection between braid groups and crystallographic groups, because from it we know explicitly the holonomy representation of the quotient group $B_{n}/[P_{n},P_{n}]$ and then applying Lemma 4 was proved that there is the short exact sequence of equation (2), and the middle group $B_{n}/[P_{n},P_{n}]$ is a crystallographic group (see [6, Proposition 1]).

Using [6, Proposition 1 and Corollary 10] it is possible to produce other crystallographic groups as follows. Let $H$ be a subgroup of $S_{{n}}$, and consider the following short exact sequence $1\longrightarrow\frac{P_{n}}{[P_{n},P_{n}]}\longrightarrow\widetilde{H}_{n}\stackrel{{\scriptstyle\overline{\sigma}}}{{\longrightarrow}}H\longrightarrow 1$ induced by the sequence given in equation (2), where $\widetilde{H}_{n}$ is defined by

[TABLE]

Then the group $\widetilde{H}_{n}$ defined by equation (3) is a crystallographic group of dimension $n(n-1)/2$ with holonomy group $H$, see [6, Corollary 13]. Was proved in [6, Theorem 2] that if $n\geq 3$ then the quotient group $B_{n}/[P_{n},P_{n}]$ has no finite-order elements of even order. Hence using the information above was proved that if $H$ is a finite $2$-group, then $H$ is the holonomy group of some flat manifold $M$. Further, the dimension of $M$ may be chosen to be $n(n-1)/2$, where $n$ is an integer for which $H$ embeds in the symmetric group $S_{{n}}$, and the fundamental group of $M$ is isomorphic to the subgroup $\widetilde{H}_{n}$ of $B_{n}/[P_{n},P_{n}]$, see [6, Theorem 20].

3 Bieberbach groups with abelian holonomy group from Artin braid groups

Let $G$ be a finite abelian group. In this section we shall prove Theorem 1 (it also reproves Theorem 6 for abelian groups using braid theory) that guaranteed that there exist $n$ and a Bieberbach subgroup $\Gamma_{G}$ of $B_{n}/[P_{n},P_{n}]$ of dimension $n(n-1)/2$ with holonomy group $G$. As a consequence $G$ is the holonomy group of a flat manifold $\mathcal{X}_{\Gamma_{G}}$ of dimension $n(n-1)/2$, where $n$ is an integer for which $G$ embeds in the symmetric group $S_{{n}}$, and the fundamental group of $\mathcal{X}_{\Gamma_{G}}$ is isomorphic to a subgroup $\Gamma_{G}$ of $B_{n}/[P_{n},P_{n}]$.

Proof of Theorem 1.

Let $q$ be the cardinality of the finite abelian group $G$, $\left|G\right|=q$.

1. (a)

We will consider 3 cases for $q$ depending of its prime decomposition.

1. (i)

Suppose that $q$ is given by product of powers of the prime 2, so $G$ is a 2-group. In this case the result follows from [6, Theorem 20]. 2. (ii)

Suppose that $q$ is even and that in the prime decomposition of $q$ appears at least one odd prime. Then, from the fundamental theorem of finite abelian groups, the group $G$ may be decomposed as $G=G_{1}\times G_{2}$, with

[TABLE]

where $p_{i}$ are odd primes not necessarily different. Let $d=\operatorname{lcm}(p_{1}^{r_{1}},p_{2}^{r_{2}},\ldots,p_{t}^{r_{t}})$ the least common multiple of the integers $p_{1}^{r_{1}},p_{2}^{r_{2}},\ldots,p_{t}^{r_{t}}$.

For $1\leq l\leq t$, let $n_{l}=\sum_{i=1}^{l}p_{i}^{r_{i}}$, and for $1\leq f\leq v$ let $m_{f}=\sum_{i=1}^{f}2^{r_{t+i}}$. So from [8, Remark] $G$ is a subgroup of $S_{{n}}$, with $n_{t}+m_{v}=n$. For convention, let $n_{0}=0$.

STEP 1. Consider the following subsets of $B_{n}/[P_{n},P_{n}]$,

[TABLE]

where

[TABLE]

for $0\leq l\leq t-1$, is the element given in [6, Equation (14)] and

[TABLE]

for $1\leq f\leq v$, where

[TABLE]

is the injective homomorphism from Theorem 7 and the element $\sigma_{1}\cdots\sigma_{2^{r_{t+f}}-1}$ is in the $(t+f)$thfactor. Recall from [6, Lemma 28] that $\delta_{n_{l},\,p_{l+1}^{r_{l+1}}}$ has order $p_{l+1}^{r_{l+1}}$ in $B_{n}/[P_{n},P_{n}]$.

Let $\Gamma$ be the subgroup of $\frac{\sigma^{-1}\left(G\right)}{[P_{n},\ P_{n}]}$ generated by $X=X_{1}\cup X_{2}\cup X_{3}\cup X_{4}$. Let $L=\Gamma\cap\operatorname{Ker}(\overline{\sigma})=\Gamma\cap\left\langle A_{i,j}\,\mid\,1\leq i<j\leq n\right\rangle$. So, $L$ is a free abelian group. We shall construct a basis for $L$.

Let

[TABLE]

We need a set of coset representatives of $L$ in $\Gamma$:

[TABLE]

From the Reidemeister-Schreier method (see [12, Appendix I]) the elements

[TABLE]

form a set of generators for $L$, with $x_{j}\in X$ and $M_{k}\in M$. Now, we analyse cases for different choices of $x_{j}\in X$.

1. Case 1.

We note that for $x_{j}\in X_{2}\cup X_{3}$, we obtain $\overline{M_{k}x_{j}}=\overline{M_{k}}=M_{k}$. Then, in these cases we obtain generators $X_{2}\cup X_{3}$. 2. Case 2.

For $x_{j}=A_{1,2}\delta_{0,n_{1}}$ and for

[TABLE]

we get $y_{j,k}=M_{k}x_{j}\overline{M_{k}x_{j}}^{-1}=(A_{1,2}\delta_{0,n_{1}})^{p_{1}^{r_{1}}}$. For other cases of $M_{k}$, still taking $x_{j}=A_{1,2}\delta_{0,n_{1}}$, we get $y_{j,k}=e$, since $\overline{M_{k}x_{j}}=M_{k}x_{j}$.

A similar argument for $x_{j}=A_{n_{l}+1,n_{l}+2}\delta_{n_{l},p_{l+1}^{r_{l+1}}}$, $1\leq l\leq t-1$, is used to show that we obtain generators $(A_{n_{l}+1,n_{l}+2}\delta_{n_{l},p_{l+1}^{r_{l+1}}})^{p_{l+1}^{r_{l+1}}}$ for $1\leq l\leq t-1$ 3. Case 3.

Now we consider the case $x_{j}=\eta_{f}$, with $f\in\left\{1,2,\ldots,v\right\}$. Let $M_{k}$ be the expression:

[TABLE]

So, in these cases we get $y_{j,k}=\eta_{f}^{2^{r_{t+f}}}$. Since $(\sigma_{1}\cdots\sigma_{2^{r_{t+f}}-1})^{2^{r_{t+f}}}=\Delta^{2}_{2^{r_{t+f}}}$ is the full twist in $\frac{B_{2^{r_{t+f}}}}{[P_{2^{r_{t+f}}},P_{2^{r_{t+f}}}]}$ then from the inclusion given in equation (9) we obtain $\eta_{f}^{2^{r_{t+f}}}=\displaystyle\prod_{n_{t}+m_{f-1}+1\leq i<j\leq n_{t}+m_{f}}A_{i,j}$ in $B_{n}/[P_{n},P_{n}]$ is a product of elements of $X_{3}$. For other cases of $M_{k}$ we get $y_{j,k}=e$.

Hence $L$ can be generated by $C_{1}\cup X_{2}\cup X_{3}$, where $X_{2}$ and $X_{3}$ was given in equation (6) and equation (7), respectively, and

[TABLE]

Let $C_{3}=X_{3}$ (see equation (7)). Let $I$ denote the index set

[TABLE]

and let

[TABLE]

Hence, the set

[TABLE]

is a basis for the free abelian group $L$.

STEP 2. Now, we prove that $\Gamma$ is torsion free. Suppose on the contrary that $w$ is a non-trivial torsion element of $\Gamma$. First we observe that, since $B_{n}/[P_{n},P_{n}]$ has no $2$-torsion [6, Theorem 2] then, in case $w$ is a non-trivial finite order element its torsion is odd. Furthermore, since $L$ is torsion free then $w\notin L$ and it projects onto an element of odd order in $G_{1}$. Recall that $d=\operatorname{lcm}(p_{1}^{r_{1}},p_{2}^{r_{2}},\ldots,p_{t}^{r_{t}})$. Hence there exists $\theta\in L$ and for $0\leq l\leq t-1$ there exist $x_{l+1}\in\left\{0,1,\ldots,p_{l+1}^{r_{l+1}}-1\right\}$ with $(x_{1},x_{2},\cdots,x_{t})\neq(0,0,\cdots,0)$ such that

[TABLE]

The elements $A_{i,j}$ with $n_{t}+1\leq i<j\leq n$ does not belong to $\theta$ because $w$ projects onto an element of odd order in $G_{1}$.

Let $A=(A_{1,2}\delta_{0,n_{1}})^{x_{1}}(A_{n_{1}+1,n_{1}+2}\delta_{n_{1},p_{2}^{r_{2}}})^{x_{2}}\cdots(A_{n_{t-1}+1,n_{t-1}+2}\delta_{n_{t-1},p_{t}^{r_{t}}})^{x_{t}}$. Recall, from the expression of $w$ given in equation (14), that $(x_{1},x_{2},\cdots,x_{t})\neq(0,0,\cdots,0)$, so at least one the $x_{i}$’s is different from zero. We will prove the case $x_{1}\neq 0$, and point out that the other cases follows in a similar way.

From $\delta_{0,p_{1}^{r_{1}}}^{p_{1}^{r_{1}}}=1$ in $B_{n}/[P_{n},P_{n}]$ we conclude that

[TABLE]

Therefore

[TABLE]

Since $\theta\in L$ then using the basis of $L$ given in equation (13) there are $\lambda_{i,j}\in\mathbb{Z}$ for $1\leq i<j\leq n$ such that $\theta$ is equal to the product

[TABLE]

Hence, for $w=\theta A$, we obtain

[TABLE]

Comparing the coefficient of $A_{1,2}$ in the expression of $w^{d}$, within the assumption $1\leavevmode\nobreak\ =\leavevmode\nobreak\ w^{d}$, see equation (15), equation (16) and equation (17), we obtain the following equality, in which the expression $\lambda_{1,2}$ appears $d$ times

[TABLE]

that is equivalent to the equality $\lambda_{1,2}+\lambda_{2,3}+\cdots+\lambda_{1,p_{1}^{r_{1}}}=-\frac{x_{1}}{p_{1}^{r_{1}}}$, which has no solution in $\mathbb{Z}$, because $p_{1}^{r_{1}}$ does not divides $x_{1}$ since $x_{1}<p_{1}^{r_{1}}$. It follows that $\Gamma$ is torsion free, and so is a Bieberbach group of dimension $\frac{n(n-1)}{2}$ with holonomy group $G\subseteq S_{{n}}$. 3. (iii)

Suppose that $q$ is odd. The proof is similar to the case given in item $(ii)$, so we just will appoint some minor details of the proof. Since $q$ is odd we may suppose that $G=G_{1}$, where $G_{1}$ was given in equation (4). Let $\Gamma$ be the subgroup of $\frac{B_{n}}{[P_{n},P_{n}]}$ generated by $X_{1}$, where $X_{1}$ is the set given in equation (5).

As in item $(ii)$ let $L=\Gamma\cap\operatorname{Ker}(\overline{\sigma})$. Then, taking the set of coset representatives $M$ of $L$ in $\Gamma$, using only the expresion given in equation (10) and using the Reidemeister-Schreier method we may conclude that a basis for $L$ is the set $A_{1}\cup A_{2}$, see equation (11) and equation (12). The proof that $\Gamma$ is torsion free is in practice the same as the one given in STEP 2 of item $(ii)$.

It follows that, in any case, given a finite abelian group $G$ there exists an integer $n$ and a Bieberbach group $\Gamma\subseteq B_{n}/[P_{n},P_{n}]$ of dimension $\frac{n(n-1)}{2}$ with holonomy group $G\subseteq S_{{n}}$. 2. (b)

Let $G$ be a finite abelian group. From item (a) there is an integer $n$ and a Bieberbach subgroup $\Gamma_{G}$ of $B_{n}/[P_{n},P_{n}]$ of dimension $n(n-1)/2$ with holonomy group $G\subseteq S_{{n}}$. By applying [2, Theorem 1] we have that there exists a manifold ${\cal X}_{\Gamma_{G}}$ with fundamental group $\Gamma_{G}$ and holonomy group $G$.

∎

Remark 9.

We note that, for any finite abelian group $G\leq S_{{n}}$, in Theorem 1 we exhibit a Bieberbach group $\Gamma_{G}\leq B_{n}/[P_{n},P_{n}]$ of dimension $n(n-1)/2$, with holonomy group $G$ and such that we may describe explicitly the holonomy representation using Proposition 8. It is very useful to have an explicit description of the holonomy representation of the Bieberbach group $\Gamma_{G}$ to deduce some geometric properties of the flat manifold ${\cal X}_{\Gamma_{G}}$ with fundamental group $\Gamma_{G}$. For instance, to see this, in Section 4 we will explore in more details the Bieberbach subgroup $\Gamma_{G}$ of $B_{n}/[P_{n},P_{n}]$ in the case in which $G$ is a cyclic group of odd order.

Let $H=\mathbb{Z}_{n_{1}}\times\mathbb{Z}_{n_{2}}\times\cdots\times\mathbb{Z}_{n_{t}}$ be a finite abelian group, where $n_{1},\ldots,n_{t}$ are powers of (not necessarily distinct) primes. Let $m$ be an integer such that $\sum_{i=1}^{t}\,n_{i}\leq m$. We may exhibit other Bieberbach groups with holonomy group $H$ in $\frac{B_{m}}{[P_{m},\ P_{m}]}$ of lower dimension than $m(m-1)/2$ as showed in the following result. Let $\binom{n}{k}$ denote the binomial coefficient.

Proposition 10.

Let $H=\mathbb{Z}_{n_{1}}\times\mathbb{Z}_{n_{2}}\times\cdots\times\mathbb{Z}_{n_{t}}$ be any finite abelian group, where $n_{1},\ldots,n_{t}$ are powers of (not necessarily distinct) primes. Let $m$ be an integer such that $\sum_{i=1}^{t}\,n_{i}\leq m$. Then there exist a Bieberbach subgroup of $\frac{B_{m}}{[P_{m},\ P_{m}]}$ of dimension $\sum_{i=1}^{t}\,\binom{n_{i}}{2}\leq\binom{m}{2}$ with holonomy group $H$.

Proof.

Let $H=\mathbb{Z}_{n_{1}}\times\mathbb{Z}_{n_{2}}\times\cdots\times\mathbb{Z}_{n_{t}}$ be a finite abelian group, where $n_{1},\ldots,n_{t}$ are powers of (not necessarily distinct) primes and let $m$ be an integer such that $\sum_{i=1}^{t}\,n_{i}\leq\leavevmode\nobreak\ m$. From Theorem 7 we have that $\overline{\iota}\colon\thinspace\displaystyle\frac{B_{n_{1}}}{[P_{n_{1}},P_{n_{1}}]}\times\cdots\times\frac{B_{n_{t}}}{[P_{n_{t}},P_{n_{t}}]}\longrightarrow\frac{B_{m}}{[P_{m},P_{m}]}$ is an injective homomorphism. Using Theorem 1 we construct a Bieberbach group $\Gamma_{\mathbb{Z}_{n_{k}}}$ in $\frac{B_{n_{k}}}{[P_{n_{k}},\ P_{n_{k}}]}$ of dimension $\binom{n_{k}}{2}$, for all $1\leq k\leq t$. Hence, $\overline{\iota}(\Gamma_{\mathbb{Z}_{n_{1}}}\times\cdots\times\Gamma_{\mathbb{Z}_{n_{t}}})\leq\frac{B_{m}}{[P_{m},\ P_{m}]}$ is a Bieberbach group of dimension $\sum_{i=1}^{t}\,\binom{n_{i}}{2}\leq\binom{m}{2}$ with holonomy group $H$. ∎

4 $\mathbb{Z}_{q}$-manifolds for $q$ an odd number

Let $q=p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots p_{t}^{r_{t}}$ be an odd number, where $\left\{p_{1},\ldots,p_{t}\right\}$ are distinct odd primes and $r_{i}\geq 1$ for all $1\leq i\leq t$. We shall consider the cyclic group $G$ of order $q$ isomorphic to $\mathbb{Z}_{p_{1}^{r_{1}}}\times\mathbb{Z}_{p_{2}^{r_{2}}}\times\cdots\times\mathbb{Z}_{p_{t}^{r_{t}}}$. For $1\leq l\leq t$, let $n_{l}=\sum_{i=1}^{l}p_{i}^{r_{i}}$. From [8, Remark] $G$ is a subgroup of $S_{{n}}$, with $n=n_{t}$. For convention, let $n_{0}=0$. Let $\Gamma_{\mathbb{Z}_{q}}\leq B_{n}/[P_{n},P_{n}]$ be the Bieberbach group constructed in Theorem 1. For short we just write $\Gamma_{q}$ to indicate the group $\Gamma_{\mathbb{Z}_{q}}$. In this section we study the existence of Anosov diffeomorphisms for flat manifolds $\mathcal{X}_{\Gamma_{q}}$ with cyclic holonomy $\mathbb{Z}_{q}$ and fundamental group $\Gamma_{q}\leq B_{n}/[P_{n},P_{n}]$, a quotient of the Artin braid group. We also give a presentation of the Bieberbach group $\Gamma_{q}$ and a set of generators of its center. The Bieberbach group $\Gamma_{q}$ fits into a short exact sequence $1\longrightarrow\mathbb{Z}^{\frac{n(n-1)}{2}}\longrightarrow\Gamma_{q}\longrightarrow\mathbb{Z}_{q}\longrightarrow 1$ and have holonomy representation

[TABLE]

induced from the action by conjugacy of $\mathbb{Z}_{q}$ over the group $\mathbb{Z}^{\frac{n(n-1)}{2}}$.

4.1 Holonomy representation as a matrix representation and proof of Theorem 2

In the proof of Theorem 1 were considered the subsets of $P_{n}/[P_{n},P_{n}]$

[TABLE]

where $I$ denote the index set

[TABLE]

From the proof of Theorem 1 we know that ${\cal B}=A_{1}\cup A_{2}$ is a basis of the free abelian group $L=\mathbb{Z}^{\frac{n(n-1)}{2}}$. For $0\leq l\leq t-1$ we consider the element $\delta_{n_{l},\,p_{l+1}^{r_{l+1}}}$ defined in equation (8). Let

[TABLE]

The element $\delta$ projects via $\overline{\sigma}\colon B_{n}/[P_{n},P_{n}]\longrightarrow S_{{n}}$ onto a generator of a cyclic subgroup (of order $q$) of $S_{{n}}$. Notice that the action of $\mathbb{Z}_{q}$ over the basis ${\cal B}$ may be computed using Proposition 8. Also note that

[TABLE]

Now we order the elements of the basis ${\cal B}$ using the action of $\mathbb{Z}_{q}$ over ${\cal B}$:

• •

For $1\leq j\leq t$ let

[TABLE]

and let

[TABLE]

for $1\leq k\leq p_{j}^{r_{j}}-1$. The element $e_{j,0,1}$ is fixed by the action for all $1\leq j\leq t$. Since $\left(A_{n_{j-1}+1,n_{j-1}+2}\right)^{q}$ does not belong to ${\cal B}$ then we rewrite it using elements of ${\cal B}$

[TABLE]

Hence, in this case, the action may be described using an arrows diagram as $e_{j,0,2}\longmapsto e_{j,0,3}\longmapsto e_{j,0,4}\longmapsto\cdots\longmapsto e_{j,0,p_{j}^{r_{j}}-1}\longmapsto e_{j,0,p_{j}^{r_{j}}}\longmapsto e_{j,0,1}^{q}\cdot\displaystyle\prod_{1\leq k\leq p_{j}^{r_{j}}-1}e_{j,0,k+1}^{-1}$ and $e_{j,0,1}\longmapsto e_{j,0,1}$, for all $1\leq j\leq t$.

• •

For each $1\leq j\leq t$ there are $\frac{p_{j}^{r_{j}}-3}{2}$ orbits of size $p_{j}^{r_{j}}$ given by

[TABLE]

for $1\leq k\leq p_{j}^{r_{j}}$ and $1\leq h\leq\frac{p_{j}^{r_{j}}-3}{2}$.

• •

For $t+1\leq j<h\leq 2t$ let

[TABLE]

for $1\leq k\leq p_{j}^{r_{j}}p_{h}^{r_{h}}$.

Consider the following (lexicographic) ordered basis of $L=\mathbb{Z}^{\frac{n(n-1)}{2}}$

[TABLE]

Let

[TABLE]

square matrices of order $z$. Therefore, using the ordered basis ${\cal B}$ of $L=\mathbb{Z}^{\frac{n(n-1)}{2}}$, we may conclude from the information above that we obtain the following matrix description of the holonomy representation of $\Gamma_{q}$ given in equation (18).

Theorem 11.

Let $\Gamma_{q}$ be the Bieberbach group constructed in Theorem 1 with holonomy group $\mathbb{Z}_{q}$ and holonomy representation $\psi_{q}\colon\mathbb{Z}_{q}\longrightarrow\operatorname{Aut}\left(\mathbb{Z}^{\frac{n(n-1)}{2}}\right)$. Let ${\cal B}$ be the ordered basis of the free Abelian group $L=\mathbb{Z}^{\frac{n(n-1)}{2}}$ given in equation (19). Then we obtain the following matrix decomposition of $\psi_{q}$:

[TABLE]

In the case $q=3$ equation (20) has the form $[\psi_{3}(\theta)]_{{\cal B}}={\cal N}_{3}[3]$. The characteristic polynomial of $[\psi_{q}(\theta)]_{{\cal B}}$ is

[TABLE]

Proof.

The matrix decomposition of $\psi_{q}$ given in equation (20) follows from the description of the action by conjugacy of $\mathbb{Z}_{q}$ over ${\cal B}$ given in the first part of this subsection. Rest to compute the characteristic polynomial of $[\psi_{q}(\theta)]_{{\cal B}}$. Removing the first line and column of the matrix ${\cal N}_{z}[q]$ we obtain the companion matrix of the monic polynomial $1+x+x^{2}+\cdots+x^{z-1}$ , so the characteristic polynomial of the matrix ${\cal N}_{z}[q]$ is $(x-1)(1+x+x^{2}+\cdots+x^{z-1})=x^{z}-1$. The matrix ${\cal M}_{z}$ is the companion matrix of the monic polynomial $x^{z}-1$. The result follows from these observations. ∎

We shall use Theorem 11 to prove Theorem 2, but first we need some definitions.

Definition.

A diffeomorphism $f\colon M\rightarrow M$ from a Riemannian manifold into itself is called Anosov and that $M$ has an hyperbolic structure if the following condition is satisfied: there exists a splitting of the tangent bundle $T(M)=E^{s}+E^{u}$ such that $Df\colon E^{s}\rightarrow E^{s}$ is contracting and $Df\colon E^{u}\rightarrow E^{u}$ is expanding.

Classification of all compact manifolds supporting Anosov diffeomorphism play an important rôle in the theory of dynamical systems, that notion represents a kind of global hyperbolic behavior, and provides examples of stable dynamical systems. The problem of classifying those compact manifolds (up to diffeomorphism) which admit an Anosov diffeomorphism was first proposed by S. Smale [15]. H. Porteous gave an interesting algebraic characterisation of the existence of Anosov diffeomorphisms for flat manifolds that just depends on the holonomy representation, see [14, Theorems 6.1 and 7.1]. Now, using the information above, we are able to prove Theorem 2.

Proof of Theorem 2.

Let $q=p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots p_{t}^{r_{t}}$ be an odd number, where $p_{i}^{r_{i}}$ are distinct odd primes and $r_{i}\geq 1$ for all $1\leq i\leq t$. From Theorem 1 let $\mathcal{X}_{\Gamma_{q}}$ be the flat manifold of dimension $\frac{n(n-1)}{2}$ with fundamental group $\Gamma_{q}\leq B_{n}/[P_{n},P_{n}]$ and holonomy group $\mathbb{Z}_{q}$.

1. (a)

Since the determinant of the matrix (representation) given in equation (20) is equal to 1 then follows from Remark 5 that the flat manifold $\mathcal{X}_{\Gamma_{q}}$ is orientable. 2. (b)

From [4, Theorems 6.4.12 and 6.4.13] we may compute the first Betti number of $\mathcal{X}_{\Gamma_{q}}$ as $\beta_{1}(\mathcal{X}_{\Gamma_{q}})=\frac{n(n-1)}{2}-\operatorname{rank}\left([\psi_{\Gamma_{q}}(\theta)]_{\cal B}-I\right)=\displaystyle\sum_{i=1}^{t}\frac{p_{i}^{r_{i}}-1}{2}+\frac{t(t-1)}{2}$ where $[\psi_{\Gamma_{q}}(\theta)]_{\cal B}$ is the matrix given in equation (20) and $I$ is the identity matrix of order $\frac{n(n-1)}{2}$. 3. (c)

Since $q$ is odd then $-1$, $i$, $-i$ are not eigenvalues of the matrix $[\psi_{q}(\theta)]_{{\cal B}}$. In Theorem 11 we computed the characteristic polynomial of $[\psi_{q}(\theta)]_{{\cal B}}$ and from it we conclude that if the numbers $1$, $\omega$, $\omega^{2}$, $-\omega$ and $-\omega^{2}$ (where $\omega^{3}=1$) are eigenvalues of the matrix $[\psi_{q}(\theta)]_{{\cal B}}$ then they have multiplicity greater than one, except in the case in which $q=3$. In the case $q=3$ the eigenvalues of the matrix $[\psi_{3}(\theta)]_{{\cal B}}$ are the cubical roots of the unity, with multiplicity one. Hence, from [14, Theorem 7.1] the flat manifold $\mathcal{X}_{\Gamma_{q}}$ admits Anosov diffeomorphism if and only if $q\neq 3$.

∎

Now we use Johnson’s method [9] to give a presentation for the group $\Gamma_{q}$ of Theorem 2.

Proposition 12.

Let $\mathbb{Z}_{q}\subseteq S_{{n}}$. The Bieberbach group $\Gamma_{q}$ fits into a short exact sequence

[TABLE]

and have a presentation given by:

1. (a)

Generators:

1. (i)

$e_{j,h,k}\in{\cal B}$* (see equation (19)),* 2. (ii)

$A_{1,2}\delta_{0,n_{1}},\,A_{n_{1}+1,n_{1}+2}\delta_{n_{1},p_{2}^{r_{2}}},\,\cdots,\,A_{n_{t-1}+1,n_{t-1}+2}\delta_{n_{t-1},p_{t}^{r_{t}}}$. 2. (b)

Relations:

1. (i)

$[e_{j,h,k},\,e_{r,u,s}]=1$, for all $e_{j,h,k},\,e_{r,u,s}\in{\cal B}$, 2. (ii)

$\left(A_{n_{j-1}+1,n_{j-1}+2}\delta_{n_{j-1},p_{j}^{r_{j}}}\right)^{p_{j}^{r_{j}}}=e_{j,0,1}$, for $1\leq j\leq t$. 3. (iii)

[TABLE]

taking $k+1(modp_{j}^{r_{j}})$ or $k+1(modp_{j}^{r_{j}}p_{h}^{r_{h}})$ in the respective case.

Proof.

The short exact sequence follows from the construction given in Theorem 1. The presentation was obtained applying the method of [9, Chapter 10]. Relations $(ii)$ follows using the action by conjugacy:

[TABLE]

The conjugacy part (relations $(iii)$) was described in the computations given before equation (19). ∎

Next, we compute the rank and give a basis of the center of $\Gamma_{q}$.

Theorem 13.

Let $\Gamma_{q}$ be the Bieberbach subgroup of $B_{n}/[P_{n},P_{n}]$ of dimension ${n}({n}-1)/2$, with holonomy group $\mathbb{Z}_{q}$. Then the center of the Bieberbach group $\Gamma_{q}$, $\mathcal{Z}(\Gamma_{q})$, is the free abelian group of rank $\displaystyle\sum_{i=1}^{t}\frac{p_{i}^{r_{i}}-1}{2}+\frac{t(t-1)}{2}$:

[TABLE]

where the elements $e_{*,*,*}$ belongs to the basis ${\cal B}$.

Proof.

The expression of $\mathcal{Z}(\Gamma_{q})$ follows from Lemma 5.2(3) of [16] that asserts that the center of $\Gamma_{q}$ is $(\mathbb{Z}^{\frac{n(n-1)}{2}})^{\mathbb{Z}_{q}}$. That means that $\mathcal{Z}(\Gamma_{q})$ is the group given by the fixed elements of the action by conjugacy of $\mathbb{Z}_{q}$ over the group $\mathbb{Z}^{\frac{n(n-1)}{2}}$. Hence the rank of $\mathcal{Z}(\Gamma_{q})$ corresponds to the number of orbits of the action by conjugacy of $\mathbb{Z}_{q}$ on the basis ${\cal B}$ given in equation (19) and a basis of $\mathcal{Z}(\Gamma_{q})$ is obtained taking the product of all the elements in each orbit. The result follows analising the action described in the computations given before equation (19) (that is equivalent to analise the fixed elements of the matrix $[\psi_{q}(\theta)]_{{\cal B}}$ given in equation (20)). ∎

Remark 14.

Notice that we may quotient $\Gamma_{q}$ by subgroups of the center of $\Gamma_{q}$ that are invariant under the action by conjugacy obtaining crystallographic groups (not necessarily torsion free). In particular, if we consider the quotient of $\Gamma_{q}$ by its center we obtain the group of inner automorphisms of $\Gamma_{q}$ (see [16, Theorem 5.1, page 65]) $\Gamma_{q}/\mathcal{Z}(\Gamma_{q})=\operatorname{Inn}(\Gamma_{q})$ that is a crystallographic group of dimension

[TABLE]

that fits into a short exact sequence $1\longrightarrow\frac{\mathbb{Z}^{\frac{n(n-1)}{2}}}{\mathcal{Z}(\Gamma_{q})}\longrightarrow\frac{\Gamma_{q}}{\mathcal{Z}(\Gamma_{q})}\stackrel{{\scriptstyle\overline{\sigma}}}{{\longrightarrow}}\mathbb{Z}_{q}\longrightarrow 1$.

4.2 $\mathbb{Z}_{p^{r}}$-manifolds with $p$ an odd prime number

Let $n=p^{r}$ where $p$ is an odd prime and $r\geq 1$. That means that $n$ is the odd number $q$ of the first paragraph of this section taking $t=1$. Let $\mathbb{Z}_{p^{r}}\subseteq S_{{n}}$ be the cyclic group generated by the permutation $\theta=(p^{r},p^{r}-1,\ldots,3,2,1)\in S_{{n}}$. Let $\Gamma_{\mathbb{Z}_{p^{r}}}\leq B_{p^{r}}/[P_{p^{r}},P_{p^{r}}]$ be the Bieberbach group constructed in Theorem 1. For short we just write $\Gamma_{p^{r}}$ to indicate the group $\Gamma_{\mathbb{Z}_{p^{r}}}$. The Bieberbach group $\Gamma_{p^{r}}$ fits into a short exact sequence $1\longrightarrow\mathbb{Z}^{\frac{p^{r}(p^{r}-1)}{2}}\longrightarrow\Gamma_{p^{r}}\longrightarrow\mathbb{Z}_{p^{r}}\longrightarrow 1$ and have holonomy representation

[TABLE]

induced from the action by conjugacy of $\mathbb{Z}_{p^{r}}$ over the group $\mathbb{Z}^{\frac{p^{r}(p^{r}-1)}{2}}$.

We recall that a Kähler manifold is a $2n$-real manifold endowed with Riemannian metric, complex structure, and a sympletic structure which are compatible at every point. A finitely presented group is a Kähler group if it is the fundamental group of a closed Kähler manifold. For more details about Kähler manifolds see [5], [10] and also [16, Section 7]. In the proof of Theorem 3 we will apply an algebraic characterisation of Kähler manifolds (see [16, Propositions 7.1 and 7.2]) in order to decide which flat manifolds of even dimension arising from Artin braid groups admit Kähler geometry.

Proof of Theorem 3.

Let $p$ be an odd prime and let $r\geq 1$. Let $\mathcal{X}_{\Gamma_{p^{r}}}$ be the flat manifold of dimension $\frac{p^{r}(p^{r}-1)}{2}$ with fundamental group $\Gamma_{p^{r}}\leq B_{p^{r}}/[P_{p^{r}},P_{p^{r}}]$ and holonomy group $\mathbb{Z}_{p^{r}}$. We need to determine in which conditions the coefficients of the real irreps that appears in the representation of $\psi_{\Gamma_{p^{r}}}$ are even, and then apply [16, Proposition 7.2] to verify if the representation is essentially complex. From equation (20) we conclude that the character vector of $\psi_{p^{r}}$ is

[TABLE]

Then the holonomy representation $\psi_{p^{r}}$ has the property that each real irrep appears $\frac{p^{r}-1}{2}$ times in its decomposition. Hence, the coefficients of the real irreps that appears in the representation of $\psi_{\Gamma_{p^{r}}}$ are even if and only if there is an integer $u\geq 1$ such that $p^{r}=4u+1$. Therefore, follows from [16, Propositions 7.1 and 7.2] that the representation given in equation (21) is essentially complex and the flat manifold $\mathcal{X}_{\Gamma_{p^{r}}}$ is Kähler if and only if there is an integer $u\geq 1$ such that $p^{r}=4u+1$. ∎

In the last 20 years, after the solution of Calabi’s conjecture by S. T. Yau, studies related to complex manifolds have grown. One of the most important classes are the so-called Calabi-Yau manifolds, that is Kähler manifolds with holonomy group contained in $SU(n)$, see [16, Section 7.2] and [5]. Follows from Theorem 2 that the flat manifold $\mathcal{X}_{\Gamma_{p^{r}}}$ of Theorem 3 is orientable, hence we obtain families of Calabi-Yau manifolds with fundamental group a Bieberbach subgroup of the quotient of the Artin braid group $B_{p^{r}}/[P_{p^{r}},P_{p^{r}}]$.

Corollary 15.

Let $p$ be an odd prime such that $p^{r}=4u+1$ for some $u,r\geq 1$. Then the flat manifold $\mathcal{X}_{\Gamma_{p^{r}}}$ of Theorem 3 is a Calabi-Yau flat manifold of dimension $2u(4u+1)$.