Almost-crystallographic subgroups of B_n/Γ₃(P_n) and infra-nilmanifolds with cyclic holonomy

Abstract

Let $n\ge 3$. In this work we study almost-crystallographic subgroups ${\tilde{H}}_{n,3}={\sigma }^{-1}(H)/{\mathrm{\Gamma }}_3(P_n)$ of $B_n/{\mathrm{\Gamma }}_3(P_n)$ with cyclic holonomy group H, where $B_n$ is the Artin braid group, $\sigma :B_n\to S_n$ is the natural projection and ${\mathrm{\Gamma }}_3(P_n)$ is the third-step element of the lower central series of the Artin pure braid group $P_n$. We study in detail the holonomy representation of ${\tilde{H}}_{n,3}$ describing it as a matrix representation. As an application, when H is a cyclic group of order a power of 2, we determine whether the infra-nilmanifold $\mathcal{X}$ with fundamental group ${\tilde{H}}_{n,3}$ is orientable and if it admits spin structure.

Introduction

If $n\in \mathbb{N}$, let $B_n$ denote the (Artin) braid group on n strings. The Artin braid group $B_n$ was introduced by Artin in 1925 [1] and further studied in 1947 [2], where a presentation for this group was given with generators ${\sigma }_1,\dots ,{\sigma }_{n-1}$ that are subject to the relations ${\sigma }_i{\sigma }_j={\sigma }_j{\sigma }_i$ for all $1\le i<j\le n-1$ for which $|i-j|\ge 2$, and ${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}$ for all $1\le i\le n-2$. Let $\sigma :B_n⟶S_n$ be the homomorphism defined on the given generators of $B_n$ by $\sigma ({\sigma }_i)=(i,i+1)$ for all $1\le i\le n-1$. The (Artin) pure braid group $P_n$ on n strings is defined to be the kernel of σ, from which we obtain the following short exact sequence:

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

If G is a group, recall that its lower central series ${\{{\mathrm{\Gamma }}_k(G)\}}_{k\in \mathbb{N}}$ is defined by ${\mathrm{\Gamma }}_1(G)=G$, and ${\mathrm{\Gamma }}_k(G)=[{\mathrm{\Gamma }}_{k-1}(G),G]$ for all $k\ge 2$ (if H and K are subgroups of G, $[H,K]$ is defined to be the subgroup of G generated by the commutators of the form $[h,k]=hk{h}^{-1}{k}^{-1}$, where $h\in H$ and $k\in K$). Note that ${\mathrm{\Gamma }}_2(G)$ is the commutator subgroup of G, and that ${\mathrm{\Gamma }}_k(G)$ is a normal subgroup of G for all $k\in \mathbb{N}$. In our setting, since $P_n$ is normal in $B_n$, it follows that ${\mathrm{\Gamma }}_k(P_n)$ is also normal in $B_n$, then equation (1) fits into the following short exact sequence

$$1⟶P_n/{\mathrm{\Gamma }}_k(P_n)⟶B_n/{\mathrm{\Gamma }}_k(P_n)\stackrel{\bar{\sigma }}{⟶}{S}_n⟶1.$$

In this paper, we continue our study of representations coming from the action by conjugation of quotients of the Artin braid group $B_n$ by elements of the lower central series ${({\mathrm{\Gamma }}_k(P_n))}_{k\in \mathbb{N}}$ of the Artin pure braid group $P_n$. In the paper [14], we analyzed the holonomy representation of Bieberbach subgroups of the crystallographic group $B_n/{\mathrm{\Gamma }}_2(P_n)$, by splitting a matrix description given by us as a direct sum of irreducible representations, and with such information we explored some properties of the flat manifolds with fundamental group those Bieberbach subgroups of $B_n/{\mathrm{\Gamma }}_2(P_n)$. Now, we consider subgroups of $B_n/{\mathrm{\Gamma }}_3(P_n)$.

Let N be any connected, simply connected and nilpotent Lie group and let us to denote by $\mathrm{Aut}\left(N\right)$ the group of automorphisms of N as a Lie group. Now, consider a maximal and compact subgroup C of $\mathrm{Aut}\left(N\right)$. It is well known that $(N⋊C)\le (N⋊\mathrm{Aut}\left(N\right))$ acts on N as follows

$$(n,\phi )\cdot m=n\phi (m),\text{ for all }n,m\in N\text{ and }\phi \in C.$$

We denote the group $N⋊\mathrm{Aut}\left(N\right)$ by $Aff(N)$ and we called it the group of affine diffeomorphisms of N.

Definition

([4, Sec. 2.2, p. 15]) An almost-crystallographic group is a discrete subgroup Π of the semi-direct product $N⋊C$ that acts properly and discontinuously on N such that $N/\mathrm{\Pi }$ is compact. If in addition Π is torsion free then Π is called an almost-Bieberbach group. The closed manifold $N/\mathrm{\Pi }$ is called an infra-nilmanifold and a nilmanifold if $\mathrm{\Pi }\subset N$. In the case that N is abelian this terminology corresponds to crystallographic group, Bieberbach group and flat Riemannian manifold, respectively.

It was proposed by Fröbenius to consider crystallographic groups up to affine isomorphisms instead up to isomorphisms, the work of Fröbenius inspired to L. Bieberbach to prove that all the isomorphisms between crystallographic groups are induced by affine conjugations. This terminology is used in this case as well, so infra-nilmanifolds are determined completely (up to an affine diffeomorphism) by their fundamental groups that are almost-Bieberbach groups. For more details about almost-crystallographic groups, almost-Bieberbach groups and infra-nilmanifolds see [3], [4], [9] and [15].

Let $n\ge 3$. In the present paper, we give a matrix description of the holonomy representation of the almost-crystallographic subgroups ${\tilde{H}}_{n,3}={}^{{\sigma }^{-1}(H)}{/}_{{\mathrm{\Gamma }}_3(P_n)}$ of ${}^{B_n}/_{{\mathrm{\Gamma }}_3(P_n)}$, where $H\le S_n$. In case that H is of order $2^t{3}^s$ then ${\tilde{H}}_{n,3}$ is an almost-Bieberbach group, see [8, Corollary 15]. We fix the case in which H is of order $2^t$ and, using the matrix description of the holonomy representation of ${\tilde{H}}_{n,3}$, we will be able to determine when the infra-nilmanifold with fundamental group ${\tilde{H}}_{n,3}$ is orientable and in these cases, if they admit spin structure.

This paper is organized as follows. In Section 2, we recall some definitions and results about almost-crystallographic groups, almost-Bieberbach groups and Artin braid groups and their quotients by the elements of the lower central series ${({\mathrm{\Gamma }}_k(P_n))}_{k\in \mathbb{N}}$ of $P_n$.

In Section 3 we are interested in the case ${\tilde{H}}_{n,3}={\sigma }^{-1}(H)/{\mathrm{\Gamma }}_3(P_n)$, when H is cyclic of order $m\le n$. The holonomy representation of ${\tilde{H}}_{n,3}$ becomes into the linear representation

$${\mathrm{\Psi }}_{{\tilde{H}}_{n,3}}:H\to \mathrm{Aut}\left(\frac{P_n}{{\mathrm{\Gamma }}_2(P_n)}\right)\oplus \mathrm{Aut}\left(\frac{{\mathrm{\Gamma }}_2(P_n)}{{\mathrm{\Gamma }}_3(P_n)}\right)$$

see equations (6) and (11). This representation is defined as ${\mathrm{\Psi }}_{{\tilde{H}}_{n,3}}={\mathrm{\Psi }}_{{\tilde{H}}_{n,3}}^1\oplus {\mathrm{\Psi }}_{{\tilde{H}}_{n,3}}^2$, where

$${\mathrm{\Psi }}_{{\tilde{H}}_{n,3}}^1:H\to \mathrm{Aut}\left(\frac{P_n}{{\mathrm{\Gamma }}_2(P_n)}\right)\text{ and }{\mathrm{\Psi }}_{{\tilde{H}}_{n,3}}^2:H\to \mathrm{Aut}\left(\frac{{\mathrm{\Gamma }}_2(P_n)}{{\mathrm{\Gamma }}_3(P_n)}\right)$$

are linear representations induced by conjugation of the quotient group $B_n/{\mathrm{\Gamma }}_3(P_n)$ on $P_n/{\mathrm{\Gamma }}_2(P_n)$ and ${\mathrm{\Gamma }}_2(P_n)/{\mathrm{\Gamma }}_3(P_n)$, respectively. We recall that we see $\mathrm{Aut}\left({\mathbb{Z}}^q\right)$ as the general linear group $GL(q,\mathbb{Z})$. In Subsection 3.1 we introduce some special matrices and explore some of their properties. These matrices are used in Subsection 3.2 in a decomposition, as a direct sum, of the holonomy representation ${\mathrm{\Psi }}_{{\tilde{H}}_{n,3}}$ of the almost-crystallographic group ${\tilde{H}}_{n,3}={}^{{\sigma }^{-1}(H)}{/}_{{\mathrm{\Gamma }}_3(P_n)}$ with cyclic holonomy group H, proving, in this way, the main result of this paper, that is the following.

Theorem 1

Let $n\ge 3$ and let $m\le n$. Let $\mathcal{A}$ and $\mathcal{B}$ the sets given in equations (12) and (13), respectively, ordered as given in Subsection 3.2. Consider the ordered set ${\mathcal{B}}^{\prime }=\mathcal{A}\cup \mathcal{B}$, where the union is ordered from left to right. Then, a matrix form of the holonomy representation given in equation (2) is

$${[{\mathit{\Psi }}_{{\tilde{H}}_{n,3}}(\mu )]}_{{\mathcal{B}}^{\prime }}={[{\mathit{\Psi }}_{{\tilde{H}}_{n,3}}^1(\mu )]}_{\mathcal{A}}\oplus {[{\mathit{\Psi }}_{{\tilde{H}}_{n,3}}^2(\mu )]}_{\mathcal{B}}$$

described in

• equation (22) and equation (16) if $gcd(m,6)=6$,

• equation (21) and equation (17) if $gcd(m,6)=3$,

• equation (22) and equation (18) if $gcd(m,6)=2$,

• equation (21) and equation (19) if $gcd(m,6)=1$.

Finally in Subsection 3.3, as an application of Theorem 1, we consider the case in which H is cyclic of order $2^t$, for $t\ge 2$, and so ${\tilde{H}}_{n,3}$ is an almost-Bieberbach group. In this case using the matrix representation of ${\mathrm{\Psi }}_{{\tilde{H}}_{n,3}}$ we may decide if the corresponding infra-nilmanifold with fundamental group ${\tilde{H}}_{n,3}$ is orientable or not, and for the cases in which it is orientable we can use Theorem 7 (that corresponds to the main theorem of [6]) in order to prove that they support Spin structure, see Theorem 9. We note that not much is known about the problem of determining Spin structures on infra-nilmanifolds but some advance was given in [6] and in this paper we use it and give a family of infra-nilmanifolds that admits Spin structures.