Linear representations of the braid groups of some manifolds ^{∗}

Valerij Bardakov

January 12, 2004

Abstract

In this paper we prove that the braid group B_{n}(S^{2}) of 2–sphere, mapping class group M(0, n) of the n–punctured 2–sphere and the braid group B_{3}(P^{2}) of the projective plane are linear. Mathematics Subject Classification: 20F28, 20F36, 20G35 Key words and phrases: braid group, braid group of manifold, linear representations.

1 Introduction

One of the most intriguing problems in the theory of braid groups is the question of whether B_{n} is linear, i. e., whether it admits a faithful representation into a group of matrices over the field. This question was formulated by W. Burau [1] in 1936 who found that n–dimensional linear representation of B_{n} which for a long time had been considered as a candidate for faithful representation. However, as it was established by J. Moody [2] in 1991 this representation is not faithful for n ≥ 9. This bound was improved for n ≥ 6 by D. Long and M. Paton [3] and to n ≥ 5 by S. Bigelow [4]. Until now, it is unknown whether the Burau representation of B_{4} is faithful.

In 1990 R. Lawrence [5] introduced a family of linear representations of B_{n}. Later D. Krammer [6] and S. Bigelow [7] proved that one of this representation is faithful. Therefore B_{n} are linear for all n ≥ 2.

Braid group B_{n} is a particular case of general construction of braid group B_{n}(M) of manifold M on n strings.

Question 1 For which manifold M and natural n is the braid group B_{n}(M) on n strings linear?

The greatest interest in this problem is connected to manifolds of dimension two. In the braid groups of these manifolds only B_{n}(S^{2}) of 2–sphere S^{2} and B_{n}(P^{2}) of projective plane P^{2} have a torsion.

In this article we will prove that the mapping class group M(0, n) of the n–punctured 2–sphere and braid group B_{n}(S^{2}) of 2–sphere are linear for all n ≥ 2. Also we will prove that braid group B_{3}(P^{2}) of projective plane is linear (for n = 1, 2 this group is finite). In all cases we will construct corresponding linear representation. The result that B_{n}(S^{2}) is linear was announced in [8]. The author awares that S. Bigelow and R. D. Budney [9] give another proof that groups B_{n}(S^{2}) and M(0, n) are linear.

∗Partially supported by the Russian Foundation for Basic Research (grant number 02–01–01118).

2 Definitions and notations

We remind you of some known facts which can be found in [10, 11, 12].

The group G is a semi direct product of A and B if there exist subgroups H and K in G such that

G = HK, A ≃ H ⊴ G, B ≃ K, H ∩ K = 1.

Semi direct product is denoted as G = A ⋋ B. If A and B have presentations

A =< a_{1}, a_{2}, …, a_{k} ∥ A_{1}, A_{2}, …, A_{p} >, B =< b_{1}, b_{2}, …, b_{l} ∥ B_{1}, B_{2}, …, B_{q} >,

then G = A ⋋ B have presentation

G =< a_{1}, a_{2}, …, a_{k}; b_{1}, b_{2}, …, b_{l} ∥ A_{1}, A_{2}, …, A_{p}; B_{1}, B_{2}, …, B_{q}; b_{i}^{−1}a_{j}b_{i} = C_{ij},
1 ≤ i ≤ l, 1 ≤ j ≤ k >,

where A_{1}, A_{2}, …, A_{p}, C_{ij} are words in alphabet {a_{1}^{±1}, a_{2}^{±1}, …, a_{k}^{±1}}, B_{1}, B_{2}, …, B_{q} are words in alphabet {b_{1}^{±1}, b_{2}^{±1}, …, b_{l}^{±1}} and elements C_{i1}, C_{i2}, …, C_{ik} generate group A for each i ∈ {1, 2, …, l}.

Let M be a manifold of dimension ≥ 2. Configuration space F_{n}(M), n ∈ ℕ, for manifold M is a set

F_{n}(M) = {(z_{1}, z_{2}, …, z_{n}) ∈ M^{n}|z_{i} ≠ z_{j} for i ≠ j}

of ordering collections of n distinct points from M. The fundamental group P_{n}(M) = π_{1}(F_{n}(M)) of the space F_{n}(M) is the pure braid group with n strings of the manifold M. Symmetric group S_{n} acts on the space F_{n}(M) by permuting the coordinates. This action is free and induce the regular covering of orbit space F_{n}(M)/S_{n} by F_{n}(M). The fundamental group B_{n}(M) = π_{1}(F_{n}(M)/S_{n}) is called the full braid group of M, or just the braid group of M. The regular covering projection F_{n}(M) ⟶ F_{n}(M)/S_{n} induces the short exact sequence

1 ⟶ P_{n}(M) ⟶ B_{n}(M) ⟶ S_{n} ⟶ 1.

If M is a closed, smooth manifold of dimension n ≥ 2, then the inclusion map F_{n}(M) ⟶ M^{n} induces a surjective homomorphism P_{n}(M) ⟶ π_{1}(M) × … × π_{1}(M) pure braid group P_{n}(M) on direct product n copies of fundamental group π_{1}(M). If dimM > 2 then this homomorphism is also injective. The braid groups of manifolds of dimension 2 represent the largest interest.

The 2–sphere S^{2} and projective plane P^{2} play a special role among the 2–dimension manifolds because the braid groups only of this manifolds have elements of finite order. If M is a closed surface different from S^{2} or P^{2}, then in the following sequence of groups

1 ⟶ P_{n}(E^{2}) ⟶ P_{n}(M) ⟶ ∏_{i=1}^{n} π_{1}(M) ⟶ 1

the kernel of each homomorphism is equal to the normal closure of the image of the previous homomorphism in the sequence. In this sequence E^{2} denotes the Euclidean plane and ∏_{i=1}^{n} π_{1}(M) denotes the direct product of n copies of π_{1}(M).

The classical braid group of Artin B_{n} is the braid group B_{n}(E^{2}) of Euclidean plane E^{2}. We will call B_{n} just a braid group.

The braid group B_{n}, n ≥ 2, with n strings can be defined as the group generated by n − 1 generators σ_{1}, σ_{2}, ..., σ_{n−1} with defining relations

σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1}, i = 1, 2, ..., n − 2,  (1)
σ_{i}σ_{j} = σ_{j}σ_{i}, |i − j| ≥ 2.  (2)

There is a homomorphism ν : B_{n} ⟶ S_{n} from the group B_{n} to the symmetric group S_{n} on n letters defined by

ν(σ_{i}) = (i, i + 1), i = 1, 2, …, n − 1.

The kernel of homomorphism ν is the pure braid group P_{n}. The group P_{n} admits a presentation with generators

a_{i,i+1} = σ_{i}^{2},
a_{ij} = σ_{j−1}σ_{j−2} … σ_{i+1}σ_{i}^{2}σ_{i+1}^{−1} … σ_{j−2}^{−1}σ_{j−1}^{−1}, i + 1 < j ≤ n.

and defining relations:

a_{ik}^{−ν}a_{kj}a_{ik}^{ν} = (a_{ij}a_{kj})^{ν} a_{kj} (a_{ij}a_{kj})^{−ν} ,
a_{km}^{−ν}a_{kj}a_{km}^{ν} = (a_{kj}a_{mj})^{ν} a_{kj} (a_{kj}a_{mj})^{−ν} ,  m < j,
a_{im}^{−ν}a_{kj}a_{im}^{ν} = [a_{ij}^{−ν}, a_{mj}^{−ν}]^{ν} a_{kj} [a_{ij}^{−ν}, a_{mj}^{−ν}]^{−ν} , i < k < m,
a_{im}^{−ν}a_{kj}a_{im}^{ν} = a_{kj},  k < i; m < j or m < k,

where ν = ±1.

The subgroup U_{n} of P_{n} which is generated by a_{1,n}, a_{2,n}, …, a_{n−1,n} is free and normal in P_{n}. The group P_{n} is a semi direct product of U_{n} and P_{n−1}. Hence the group P_{n} is a semi direct product

P_{n} = U_{n} ⋋ (U_{n−1} ⋋ (… ⋋ (U_{3} ⋋ U_{2})) …),

where U_{i} =< a_{1,i}, a_{2,i}, …, a_{i−1,i} >, i = 2, 3, …, n, is a free group of rank i − 1.

We remind you of the definition (see [5, 6, 7]) of faithful linear representation of Lawrence– Krammer braid group B_{n}. Let V_{n} be a free module dimension m = n(n − 1)/2 with basis v_{ij}, 1 ≤ i < j ≤ n, over ring ℤ[t^{±1}, q^{±1}] of Laurent polynomials on two variables. Then representation ρ : B_{n} ⟶ GL(V_{n}) is defined by action by σ_{i}, i = 1, 2, …, n − 1, on module V_{n} by equalities (we will write σ_{i}(v_{j,k}) or v_{j,k}^{σ_{i}} instead of ρ(σ_{i})(v_{j,k}) or v_{j,k}^{ρ(σ_{i})} accordingly):

σ_{i}(v_{k,i}) = (1 − q)v_{k,i} + qv_{k,i+1} + q(q − 1)v_{i,i+1},
σ_{i}(v_{k,i+1}) = v_{k,i} if k < i,
σ_{i}(v_{i,i+1}) = tq^{2}v_{i,i+1},
σ_{i}(v_{i,l}) = tq(q − 1)v_{i,i+1} + (1 − q)v_{i,l} + qv_{i+1,l} if i + 1 < l,
σ_{i}(v_{i+1,l}) = v_{i,l},
σ_{i}(v_{k,l}) = v_{k,l} if {k, l} ⋂{i, i + 1} = ∅.

3 Linear representations

As it was proved (see [11, theorem 1.11; 13]) the braid group B_{n}(S^{2}) of the 2–sphere S^{2} admits a presentation with generators δ_{1}, δ_{2}, …, δ_{n−1} and defining relations:

δ_{i}δ_{i+1}δ_{i} = δ_{i+1}δ_{i}δ_{i+1} if i = 1, 2, ..., n − 2,
δ_{i}δ_{j} = δ_{j}δ_{i} if |i − j| ≥ 2,
δ_{1}δ_{2} … δ_{n−2}δ_{n−1}^{2}δ_{n−2} … δ_{2}δ_{1} = 1.

¿From this relations we see that B_{n}(S^{2}) is a homomorphic image of B_{n}. Since B_{2}(S^{2}) is a cyclic group of order 2 and B_{3}(S^{2}) is a metacyclic group of order 12, then we will consider n > 3.

R. Gillette and J. Van Buskirk [13] have studied the structure of B_{n}(S^{2}). We recall some of their results. Let

a_{i,i} = 1, a_{i,j} = δ_{i}^{−1}δ_{i+1}^{−1} … δ_{j−2}^{−1}δ_{j−1}^{2}δ_{j−2} … δ_{i+1}δ_{i}, 1 ≤ i < j ≤ n.  (1)

These elements generate the pure braid group P_{n}(S^{2}) which is a kernel of homomorphism ν : B_{n}(S^{2}) ⟶ S_{n} and ν send δ_{i} in transposition (i, i + 1), 1 ≤ i ≤ n − 1, from S_{n}. Let define the subgroup A_{n−i+1} =< a_{i,i+1}, a_{i,i+2}, …, a_{i,n} > for each i = 1, 2, …, n − 1. Subgroup A_{n} is normal in P_{n}(S^{2}) and we have the short exact sequence

1 ⟶ A_{n} ⟶ P_{n}(S^{2}) ⟶ P_{n−1}(S^{2}) ⟶ 1

and P_{n}(S^{2}) is a semi direct product: P_{n}(S^{2}) = A_{n} ⋋P_{n−1}(S^{2}). The generators of A_{n} are connected by the relation

a_{1,2}a_{1,3} … a_{1,n} = 1.

Since the following relations are held in P_{n}(S^{2})

a_{i,i+1}a_{i,i+2} … a_{i,i+n−1} = 1, i = 1, 2, …, n − 1,

where a_{j,i} = a_{i,j} if j > i and indices are taken modn, then

a_{i,n} = a_{i,n−1}^{−1} … a_{i,i+1}^{−1}a_{i−1,i}^{−1} … a_{1,i}^{−1}, i = 1, 2, …, n − 1.

Using these formulas we can exclude a_{1,n}, a_{2,n}, …, a_{n−1,n} from the set of generators of P_{n}(S^{2}). Hence A_{n−i+1} is freely generated by a_{i,i+1}, a_{i,i+2}, …, a_{i,n−1}.

The center of B_{n}(S^{2}) is the cyclic group of order 2 generated by the Dirac braid

Δ_{n} = (δ_{1}δ_{2} … δ_{n−2})^{n−1} = (a_{1,2}a_{1,3} … a_{1,n−1})(a_{2,3}a_{2,4} … a_{2,n−1}) … (a_{n−2,n−1}).

The group P_{n}(S^{2}) was splitted in the semi direct product

P_{n}(S^{2}) = A_{n} ⋋ (A_{n−1} ⋋ (… ⋋ A_{3}) …).

Let L_{n} = A_{n} ⋋ (A_{n−1} ⋋ (… ⋋ A_{4}) …). Since A_{3} is a cyclic group generated by a_{n−2,n−1}, then P_{n}(S^{2}) = L_{n}× < Δ_{n} >≃ L_{n}×ℤ_{2}. Group L_{n} is isomorphic to subgroup U_{n}⋋(U_{n−1}⋋(…⋋U_{4}) …) ≤ P_{n}.

The braid group of sphere closed relates with the group M(0, n) of mapping classes of the n–punctured 2–sphere. Presentation of the M(0, n) is derived from that of B_{n}(S^{2}) by adding the relation: Δ_{n} = (δ_{1}δ_{2} … δ_{n−2})^{n−1} = 1. There is an epimorphism from B_{n}(S^{2}) to M(0, n). The kernel of this epimorphism is equal to the center of B_{n}(S^{2}). A mapping class of the n–punctured 2–sphere can be visualized as an n–string braid between concentric 2–spheres, where the inner sphere is free to execute full revolutions.

For M(0, n) there is a short exact sequence

1 ⟶ L_{n} ⟶ M(0, n) ⟶ S_{n} ⟶ 1,

where L_{n} is a kernel of epimorphism ν : M(0, n) ⟶ S_{n} that map δ_{i} to the transposition (i, i + 1), i = 1, 2, …, n − 2.

A. I. Malcev [14, lemma 1; 10] has proved that if H is a subgroup of finite index in group G and H is linear, then G is linear too. Recall his construction. Let |G : H| = m and ψ : H ⟶ GL_{l}(F) is a faithful linear representation of H by matrices of order l over field F. In this case G is a union of right cosets

G = He ⋃ Hg_{2} ⋃ … ⋃ Hg_{m}.

For each g ∈ G we can write the product g_{i}g uniquely in the form h_{i}g_{n_{i}}, h_{i} ∈ H. Therefore for each g we have the sequence h_{1}, h_{2}, …, h_{m} from H and the sequence numbers n_{1}, n_{2}, …, n_{m}. For the sequence {n_{i}} we construct the matrix D(n_{i}) with interger coefficients by the rule:

D(n_{i}) =∥ d_{j,k} ∥∈ M_{lm}(ℤ), d_{j,n_{j}} = E_{l}, d_{j,k} = 0 if k ≠ n_{j},

where E_{l} is identical matrix of order l. Then we can map g ∈ G to the matrix

diag(ψ(h_{1}), ψ(h_{2}), …, ψ(h_{m}))D(n_{i}).

Hence we constructed the linear representation of G in GL_{lm}(F). This linear representation is faithful. Using this construction we will prove Theorem 3.1 Let R = ℤ[t^{±1}, q^{±1}] be a ring of Laurent polynomials on two variables. Then the groups M(0, n) and B_{n}(S^{2}) are linear for each n ≥ 2. For each n ≥ 4 there are inclusion maps φ : M(0, n) ⟶ GL_{m}(R) and φ_{1} : B_{n}(S^{2}) ⟶ GL_{m_{1}}(R), where m = (n−1)(n−2)n!/2, m_{1} = 2m.

Proof: As we noted above, if n = 2, 3 groups M(0, n), B_{n}(S^{2}) are finite and hence are linear. Since L_{n} is isomorphic to subgroup of B_{n}, then is a faithful linear representation ρ : L_{n} ⟶ GL_{l}(R) for l = (n − 1)(n − 2)/2. This representation is induced by Lawrence–Krammer representation of B_{n}.

Let m_{1}, m_{2}, …, m_{n!} be coset representatives of M(0, n) by subgroup L_{n}. Since M(0, n) is generated by δ_{1}, δ_{2}, …, δ_{n−1} then we can define φ only on these elements. Each generator δ_{k} acts on set of coset representatives by permutation. We find

m_{i}δ_{k} = h_{i}^{k}m_{π_{k}(i)}, i = 1, 2, …, n!, k = 1, 2, …, n − 1,

where symbol k in the upper part is an index but not an exponent (this rule will be true to the end of this proof). We compare δ_{k} with the matrix

φ(δ_{k}) = diag(ρ(h_{1}^{k}), ρ(h_{2}^{k}), …, ρ(h_{n!}^{k}))π(δ_{k}) ∈ GL_{m}(R),

where diag(ρ(h_{1}^{k}), ρ(h_{2}^{k}), …, ρ(h_{n!}^{k})) is a block–diagonal matrix; π(δ_{k}) is a block–monomial matrix in which the block on position (j, π(j)) is an identical matrix of order l, but the block on position (j, s) for s ≠ π(j) is a null matrix of order l. This linear representation φ is a faithful representation of M(0, n).

Consider the group B_{n}(S^{2}). This group contains the linear subgroup L_{n} index 2n!. As a set of right coset representatives of B_{n}(S^{2}) by subgroup L_{n} we take elements m_{i}Δ_{n}^{ϵ}, i = 1, 2, …, n!, ϵ = 0, 1, where Δ_{n} is the generator of center of B_{n}(S^{2}). Since B_{n}(S^{2}) is generated by δ_{1}, δ_{2}, …, δ_{n−1} then it is enough to define representation φ_{1} on these elements. Let us order the coset representatives m_{i}Δ_{n}^{ϵ} and denote them by n_{1}, n_{2}, …, n_{2n!}. Each generator δ_{k} of B_{n}(S^{2}) acts on these representatives by the rules

n_{j}δ_{k} = g_{j}^{k}n_{π_{k}(j)}, j = 1, 2, …, 2n!, k = 1, 2, …, n − 1.

We compare δ_{k} with the matrix

φ_{1}(δ_{k}) = diag(ρ(g_{1}^{k}), ρ(g_{2}^{k}), …, ρ(g_{2n!}^{k}))π(δ_{k}) ∈ GL_{2m}(R),

where diag(ρ(g_{1}^{k}), ρ(g_{2}^{k}), …, ρ(g_{2n!}^{k})) is a block–diagonal matrix and π(δ_{k}) is a block–monomial matrix in which the block on position (j, π(j)) is an identical matrix of order l but the block on position (j, s) for s ≠ π(j) is a null matrix of order l. This linear representation φ_{1} is a faithful representation of B_{n}(S^{2}) over ring R. Since R is included in the field of complex numbers ℂ (it is enough to take numbers t and q which are non–zero transcendental over ℚ) then we obtain the required assertions. □

Let P^{2} be a projective plane. The braid group B_{n}(P^{2}), n ≥ 1, admits a presentation [15] with generators δ_{1}, δ_{2}, …, δ_{n−1}, ρ_{1}, ρ_{2}, …, ρ_{n} and defining relations:

δ_{i}δ_{i+1}δ_{i} = δ_{i+1}δ_{i}δ_{i+1} if i = 1, 2, ..., n − 2,
δ_{i}δ_{j} = δ_{j}δ_{i} if |i − j| ≥ 2,
δ_{i}ρ_{j} = ρ_{j}δ_{i}, j ≠ i, i + 1,
ρ_{i} = δ_{i}ρ_{i+1}δ_{i} if i = 1, 2, ..., n − 1,
ρ_{i+1}^{−1}ρ_{i}^{−1}ρ_{i+1}ρ_{i} = δ_{i}^{2} if i = 1, 2, ..., n − 1,
δ_{1}δ_{2} … δ_{n−2}δ_{n−1}^{2}δ_{n−2} … δ_{2}δ_{1} = ρ_{1}^{2}.

It is clear that the map taking δ_{i} onto the transposition (i, i+1), i = 1, 2, …, n−1 and each ρ_{j} onto the identity is homomorphism of B_{n}(P^{2}) onto the symmetric group S_{n}, with kernel P_{n}(P^{2}).

The group B_{1}(P^{2}) is a cyclic group of order 2, B_{2}(P^{2}) is a finite group of order 16. For n ≥ 3 the group B_{n}(P^{2}) is infinite. Consider the case n = 3. The group P_{3}(P^{2}) contains a free subgroup A_{3}(P^{2}) =< ρ_{1}, a_{2}, a_{3} ∥ a_{2}a_{3} = ρ_{1}^{2} >, where a_{2} = δ_{1}^{2}, a_{3} = δ_{1}^{−1}δ_{2}^{2}δ_{1}. The factor group P_{3}(P^{2})/A_{3}(P^{2}) ≃ P_{2}(P^{2}) is a quaternion group of order 8. The set

{1, δ_{1}, δ_{2}, δ_{2}δ_{1}, δ_{1}δ_{2}, δ_{1}δ_{2}δ_{1}}

is a right coset representatives of B_{3}(P^{2}) by P_{3}(P^{2}). Note that P_{2}(P^{2}) has a presentation

P_{2}(P^{2}) =< ρ_{2}, ρ_{3} ∥ ρ_{2}^{2} = ρ_{3}^{2} = (ρ_{3}ρ_{2})^{2} > .

In P_{3}(P^{2}) the following equations are true:

ρ_{3}^{−ϵ}a_{2}ρ_{3}^{ϵ} = a_{2}, ϵ = ±1, ρ_{j}^{−1}ρ_{1}ρ_{j} = ρ_{1}a_{j}^{−1}, j = 2, 3,
ρ_{2}^{−1}a_{3}ρ_{2} = a_{2}a_{3}a_{2}^{−1}, ρ_{2}^{−1}a_{2}ρ_{2} = ρ_{1}a_{2}^{−1}ρ_{1}^{−1}.

¿From defining relations of B_{3}(P^{2}) we get δ_{1}, δ_{2}.

Lemma 3.1 In B_{3}(P^{2}) the following rules of conjugating are true

a_{2}^{δ_{1}} = a_{2}, ρ_{1}^{δ_{1}} = ρ_{2}a_{2}, ρ_{2}^{δ_{1}} = a_{2}^{−1}ρ_{1}, ρ_{3}^{δ_{1}} = ρ_{3},
a_{2}^{δ_{2}} = ρ_{1}^{2}a_{2}^{−1}, ρ_{1}^{δ_{2}} = ρ_{1}, ρ_{2}^{δ_{2}} = ρ_{3}δ_{2}^{2}, ρ_{3}^{δ_{2}} = δ_{2}^{−2}ρ_{2}.

Since A_{3}(P^{2}) is a free group with free generators ρ_{1}, a_{2} then using Sanov’s representation [10] we can include A_{3}(P^{2}) in SL_{2}(ℤ) assuming

\overline{ρ}_{1} = ( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) , \overline{a}_{2} = ( \begin{matrix}1 & 0 \\ 2 & 1 \end{matrix} ) ,

as images of ρ_{1} and a_{2} accordingly.

The group P_{3}(P^{2}) is an extension of A_{3}(P^{2}) by the group of quaternion of order 8. Let us chose the set

{e, ρ_{2}, ρ_{2}^{2}, ρ_{2}^{3}, ρ_{3}, ρ_{3}ρ_{2}, ρ_{3}ρ_{2}^{2}, ρ_{3}ρ_{2}^{3}}

as the set of coset representatives of P_{3}(P^{2}) by A_{3}(P^{2}). Thus free group A_{3}(P^{2}) has index 48 in B_{3}(P^{2}). Using Malcev’s construction we get

Theorem 3.2 Braid group B_{3}(P^{2}) of projective plane P^{2} is included in SL_{96}(ℤ).

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Author address:

Valerij Bardakov Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

bardakov@math.nsc.ru