Presentations of Schur covers of braid groups

1 Introduction

For a group 𝐺, consider a central group extension

of 𝐺 by the integral second homology H2⁢(G) of 𝐺. If the cohomology class u∈H2⁢(G,H2⁢(G)) corresponding to this central extension is mapped to idH2⁢(G) by the homomorphism κ:H2⁢(G,H2⁢(G))→HomZ⁢(H2⁢(G),H2⁢(G)) coming from the universal coefficient theorem, the group 𝐸 is called a Schur cover of 𝐺. A concept of Schur covers was originally introduced by Schur in a series of his works [9, 10] in the study of projective representations of finite groups. In particular, he determined the Schur cover of the symmetric groups and gave a finite presentation for it. (See Theorem 2.2.)

For n≥1, let Bn be the braid group of 𝑛-strands. The group Bn was studied in his original works by Artin [1] who gave its first finite presentation. Let Sn be the symmetric group of degree 𝑛. There is a natural surjective homomorphism from Bn to Sn. In other words, Bn is an extension of Sn. From the viewpoint of Artin groups, Sn is the Coxeter group of Bn. Now the study of the braid groups has achieved good progress and has a broad range of remarkable results by many authors. For the study of the Schur covers of Bn, it turns out that there is a unique non-trivial central extension of Bn by H2⁢(Bn) up to isomorphism from the universal coefficient theorem. (For details, see Section 2.2.) Namely, the Schur cover of Bn is essentially unique; denote it by Cn. On the other hand, Huebschmann [5] studied the homomorphism ϖ:Cn→Bn from a crossed module viewpoint and showed that it is a free Bn-crossed module generated by a single generator of Bn. Thus the group Cn has two interesting features. One is the non-trivial central extension, and the other is being a free crossed module.

Hitherto, no explicit presentation of the Schur cover of Bn has been obtained. In the main theorem of this paper, we give it as follows.

In order to show this, we use 2-cocycles of Bn and combinatorial group theory. In Section A, as an appendix, we give explicit constructions of generators of H2⁢(Sn,Z/2⁢Z)≅(Z/2⁢Z)2 for n≥4. These cocycles are used to construct those of Bn. This paper is based on the master’s thesis [8] of the second author at Hokkaido University in 2022.

2 Schur covers

First of all, we review and fix notation for Artin’s braid groups and related groups. For n≥1, let Bn be the braid group of 𝑛 strands. The groups Bn were introduced by Artin in 1925, and Artin gave the finite presentations for them in 1947. More precisely, Artin [1] obtained the first finite presentation of Bn as follows:

By the universal mapping property, there is unique homomorphism onto Sn. It is also well known that Sn has the following finite presentation:

Thus we see that there is the standard surjective homomorphism π:Bn→Sn such that π⁢(σi)=si for 1≤i≤n−1.

Here, for a group 𝐺, we review the correspondence between central extensions and 2-cocycles of 𝐺. Let 𝐴 be an abelian group and

a central extension of 𝐺 by 𝐴. Namely, i⁢(A) is contained in the center of 𝐸. Choose a set-theoretic section s:G→E of 𝜛 such that s⁢(1)=1. Further, let fE:G×G→A be the map defined by (g,h)↦s⁢(g)⁢s⁢(h)⁢(s⁢(g⁢h))−1. Then fE is a normalized 2-cocycle of 𝐺 to 𝐴.

Conversely, if a normalized 2-cocycle f:G×G→A of 𝐺 is given, then 𝑓 defines a central extension of 𝐺 by 𝐴 as follows. Let Ef be the set A×G equipped with the product defined by (a,g)⁢(b,h)=(a+b+f⁢(g,h),g⁢h). From the 2-cocycle condition of 𝑓, Ef becomes a group. Let E⁢(G,A) be the set of equivalence classes of central extensions of 𝐺 by 𝐴. It is well known that the correspondence [E]↦[fE] defines a bijection E⁢(G,A)→H2⁢(G,A), whose inverse is given by [f]↦[Ef].

We review the Schur covers of the symmetric groups and the braid groups. Here, for the simplicity, for a group 𝐺, we will abbreviate the integral homology group Hp⁢(G,Z) of 𝐺 as Hp⁢(G). For any p≥1 and any trivial 𝐺-module 𝑀, let us consider the universal coefficient theorem

We remark that a Schur cover is not unique in general. If 𝐺 is a perfect group, there exists a unique Schur cover up to isomorphism. Then (2.3) is called the universal central extension of 𝐺.

2.1 Schur covers of the symmetric groups

Here we review Schur covers of the symmetric group Sn. For any n≥4, it is known that

H1⁢(Sn)≅Z/2⁢Z,H2⁢(Sn)≅Z/2⁢Z.

Hence a Schur cover 𝐸 of Sn satisfies

(2.4)0→Z/2⁢Z→E→Sn→1.

By using the universal coefficient theorem and

HomZ⁢(H2⁢(Sn),Z/2⁢Z)≅Z/2⁢Z,ExtZ1⁢(H1⁢(Sn),Z/2⁢Z)≅Z/2⁢Z,

we have H2⁢(Sn,Z/2⁢Z)≅Z/2⁢Z⊕Z/2⁢Z. Hence there exist four non-equivalent central extensions which satisfy (2.4). As HomZ⁢(H2⁢(Sn),H2⁢(Sn))≅Z/2⁢Z consists of two elements, which are the identity map idH2⁢(Sn) and the zero map, there exist two elements of H2⁢(Sn,H2⁢(Sn)) which are mapped to idH2⁢(Sn). Each of them induces the extension (2.4) and the following commutative diagram:

Therefore, there exist two Schur covers of Sn. The 2-cocycles which correspond to the two Schur covers are given by the following theorem.

Theorem 2.2

Theorem 2.2 (Schur [9, 10] (see also [2]))

Consider Z/2⁢Z as a multiplicative group, and let 𝑧 be its generator.

1. For any α,β∈Z/2⁢Z={1,z}, there exists the 2-cocycle

   τ[α,β]:Sn×Sn→Z/2⁢Z

   such that

   τ[α,β]⁢(si,si)=α,τ[α,β]⁢(sk,sl)={1(k<l),β(l<k).

   Then we have H2⁢(Sn,Z/2⁢Z)={[τ[α,β]]∣α,β∈{1,z}}.

2. Let

   0→Z/2⁢Z→ιE[α,β]→ϖSn→1

   be the central extension corresponding to [τ[α,β]]. Then Eτ[α,β] has a presentation whose generators are t1,…,tn−1,z subject to relations

   ti2=α,z2=1,z⁢ti=ti⁢z,tj⁢tj+1⁢tj=tj+1⁢tj⁢tj+1,tk⁢tl=β⁢tl⁢tk(1≤i≤n−1, 1≤j≤n−2, 1≤k<l−1≤n−2).

   Here ι⁢(z)=z, ϖ⁢(ti)=si and ϖ⁢(z)=1.

3. If β=z, then for any α∈Z/2⁢Z, the central extension E[α,β] is a Schur cover of Sn.

In Appendix A, we give explicit calculations to obtain the 2-cocycles τ[α,β] by using combinatorial group theory since there is no explicit calculations in [9].

2.2 Schur covers of braid groups

Here we consider a Schur cover of the braid group Bn. For any n≥4, it is well known that

H1⁢(Bn)≅Z,H2⁢(Bn)≅Z/2⁢Z.

A Schur cover of Bn satisfies

(2.5)0→Z/2⁢Z→E→ϖBn→1.

By using the universal coefficient theorem and

Hom⁢(H2⁢(Bn),Z/2⁢Z)≅Z/2⁢Z,ExtZ1⁢(H1⁢(Bn),Z/2⁢Z)=0,

we have H2⁢(Bn,Z/2⁢Z)≅Z/2⁢Z. In this case, there are two non-equivalent central extensions of (2.5). Furthermore, since

κ:H2⁢(Bn,Z/2⁢Z)→Hom⁢(H2⁢(Bn),Z/2⁢Z)

is an isomorphism, the non-trivial central extension of Bn by Z/2⁢Z is the unique Schur cover. For the standard surjection π:Bn→Sn, consider the four 2-cocycles μ[α,β]:=τ[α,β]∘(π×π):Bn×Bn→Z/2⁢Z of Bn, and the following commutative diagram induced from the universal coefficient theorem:

For any α,β∈Z/2⁢Z={1,z}, let

0→Z/2⁢Z→C[α,β]→Bn→1

be a central extension corresponding to μ[α,β]. We obtain the following proposition.

Proposition 2.3

In the notation above,

1. if β=1, then the central extension is trivial;

2. C[1,z] and C[z,z] are equivalent central extensions of Bn, and each of them is a Schur cover of Bn.

3 Presentations of extensions of groups

In this section, according to [6], we review a method to obtain a presentation for an extension of presented groups. Let

be a group extension, and assume that presentations for 𝐾 and 𝑁 are given by

respectively. Set X*:={x*∣x∈X} and Y*:={y*∣y∈Y}. Let 𝐹 be the free group on X*∪Y*, and set

For any x∈X, take x′∈q−1⁢(x), and set X′:={x′∈G∣x∈X}. Let φ:F→G be the surjective homomorphism defined by x*↦x′ and y*↦ι⁢(y) for any x*∈X*, y*∈Y*. For any r=x1f1⁢⋯⁢xlfl∈R, set r*:=(x1*)f1⁢⋯⁢(xl*)fl∈F. Then

Hence we can write φ⁢(r*)=yr,1e1⁢⋯⁢yr,kek for some yr,i∈Y and ei=±1. Set vr:=(yr,1*)e1⁢⋯⁢(yr,k*)ek∈F, and R*:={r*⁢vr−1∈F∣r∈R}. Since Im⁢(ι) is the normal subgroup of 𝐺, we see that φ⁢((x*)−1⁢y*⁢x*)∈Im⁢(ι) for any x*∈X* and y*∈Y*. Thus we can write φ⁢((x*)−1⁢y*⁢x*)=ι⁢(yx,y,1a1⁢⋯⁢yx,y,kak) for some yx,y,i∈Y and ai=±1. Set

For a proof, see [6].

4 Presentations of Schur covers of braid groups

In this section, we prove our main theorem. By Theorem 2.2, for any

the 2-cocycle μ[α,β]:Bn×Bn→Z/2⁢Z defined in Section 2.2 satisfies the following equations:

Let

be the central extension corresponding to μ[α,β]. The group C[α,β] can be identified with Z/2⁢Z×Bn as a set. Under this identification, the group operation on C[α,β] is represented as

and the homomorphisms ι:Z/2⁢Z→C[α,β], ϖ:C[α,β]→Bn are represented as ι⁢(s)=(s,1), ϖ⁢(x,σ)=σ. Furthermore, there exists a standard set-theoretic section Bn→C[α,β], σ↦(1,σ). Consider the presentation ⟨z∣z2⟩ of Z/2⁢Z and Artin’s presentation of Bn as mentioned in (2.1). For simplicity, we write the above presentations as Z/2=⟨Y∣S⟩, Bn=⟨X∣R⟩. Let F⁢(X) be the free group on 𝑋, and φ:F⁢(X)→Bn the standard surjection. Since φ|X:X→Bn is injective, we regard 𝑋 as a subset of Bn through 𝜑, and we write φ⁢(x) as 𝑥 for any x∈X by abuse of language. In a similar way, we consider Y⊂Z/2⁢Z.

We use Theorem 3.1 to obtain a presentation of C[α,β]. First, set

Let 𝐹 be the free group F⁢(X*∪Y*) on X*∪Y*. Here S*={(z*)2}⊂F. Set X′:={(1,σi)∣1≤i≤n−1}. The standard homomorphism ψ:F→C[α,β] is defined by z*↦z and σi*↦(1,σi). For each i,j such that 1≤i<j≤n−1, define ri⁢j∈R and ri⁢j*∈F by

Then we have

Consider the element (1,σi)⁢(1,σj)⁢(1,σi)−1⁢(1,σj)−1 for i<j−1. By using the equations

we obtain

Next, we consider the element

Similarly, we have

From Lemma 4.1,

Set

We have φ⁢((σi*)−1⁢z*⁢σi*)∈Im⁢(ι) for any σi*∈X* and z*∈Y*, and

Hence the set T* of Theorem 3.1 is

From the above calculations and Theorem 3.1, we obtain the following theorem.

From Proposition 2.3, we have the following corollary.