THE VIRTUAL AND UNIVERSAL BRAIDS

VALERIJ G. BARDAKOV

Abstract. We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi–direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi–direct product of free groups. From these results we obtain a normal form of words in the virtual braid group. We introduce the concept of a universal braid group. This group contains the classical braid group and has as its quotient groups the singular braid group, virtual braid group, welded braid group, and classical braid group.

Recently some generalizations of classical knots and links were defined and studied: singular links [20, 5], virtual links [15, 12] and welded links [10].

One of the ways to study classical links is to study the braid group. Singular braids [1, 5], virtual braids [15, 21], welded braids [10] were defined similar to the classical braid group. A theorem of A. A. Markov [4, Ch. 2.2] reduces the problem of classification of links to some algebraic problems of the theory of braid groups. These problems include the word problem and the conjugacy problem. There are generalizations of Markov’s theorem for singular links [11], virtual links, and welded links [14].

There are some different ways to solve the word problem for the singular braid monoid and singular braid group [8, 7, 22]. The solution of the word problem for the welded braid group follows from the fact that this group is a subgroup of the automorphism group of the free group [10]. A normal form of words in the welded braid group was constructed in [13].

In this paper we study the structure of the virtual braid group V B_{n}. This is similar to the classical braid group B_{n} and welded braid group W B_{n}. The group V B_{n} contains the normal subgroup V P_{n} which is called the virtual pure braid group. The quotient group V B_{n}/V P_{n} is isomorphic to the symmetric group S_{n}. In the article we find generators and defining relations of V P_{n}. Since V B_{n} is a semi–direct product of V P_{n} and S_{n}, we should study the structure of V P_{n}. It will be proved that V P_{n} is representable as the following semi–direct product

Date: September 14, 2004.

1991 Mathematics Subject Classification. Primary 20F36; Secondary 20F05, 20F10.

Key words and phrases. Knot theory, singular knot, virtual knot, braid group, singular braid monoid, free groups, automorphism, word problem.

Authors were supported in part by the Russian Foundation for Basic Research (grant 02–01–01118).

V P_{n} = V_{n−1}^{∗} ⋊ V P_{n−1} = V_{n−1}^{∗} ⋊ (V_{n−2}^{∗} ⋊ (… ⋊ (V_{2}^{∗} ⋊ V_{1}^{∗})) …),

where V_{i}^{∗} is some (in general infinitely generated for i > 1) free subgroup of V P_{n}. From this result it follows that there exists a normal form of words in V B_{n}.

In the last section we define the universal braid group UB_{n} which contains the braid group B_{n} and has as its quotient groups the singular braid group SG_{n}, the virtual braid group V B_{n}, the welded braid group W B_{n}, and the braid group B_{n}. It is known [10] that V B_{n} has as its quotient the group W B_{n}. It will be proved that the quotient homomorphism maps V P_{n} into the welded pure braid group W P_{n}. This homomorphism agrees with the decomposition of this group into the semi-direct product given by Theorem 2 and by [2, 3].

By Artin’s theorem, B_{n} is embedded into the automorphism group Aut(F_{n}) of the free group F_{n}. In [10] it was proved that W B_{n} is also embedded into Aut(F_{n}). It is not known if it is true that SG_{n} and V B_{n} are embedded into Aut(F_{n}).

Acknowledgments. I am very grateful to Joanna Kania-Bartoszyn-ska, Jozef Przytycki, Pawel Traczyk, and Bronislaw Wajnryb for organizing the Mini–semester on Knot Theory (Poland, July, 2003) and for the invitation to participate in this very interesting and very well– organized Mini–semester. I would also like to thank Vladimir Vershinin and Andrei Vesnin for their interest to this work. Special thanks goes to the participants of the seminar “Evariste Galois” at Novosibirsk State University for their kind attention to my work.

1. different classes of braids and their properties

In this section we remind (see references from the introduction) some known facts about braid groups, singular braid monoids, virtual braid groups and welded braid groups.

1.1. The braid group and the group of conjugating automorphisms. The braid group B_{n}, n ≥ 2, on n strings can be defined as a group generated by σ_{1}, σ_{2}, …, σ_{n−1} (see Fig. 1)

Figure 1. Geometric braids representing σ_{i} and σ_{i}^{−1}

with the defining relations

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

There exists a homomorphism of B_{n} onto the symmetric group S_{n} on n letters. This homomorphism maps σ_{i} to the transposition (i, i + 1), i = 1, 2, …, n − 1. The kernel of this homomorphism is called the pure braid group and denoted by P_{n}. The group P_{n} is generated by a_{ij}, 1 ≤ i < j ≤ n (see Fig. 2). These generators can be expressed by the generators of B_{n} as follows

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.

The group P_{n} is the semi–direct product of the normal subgroup U_{n} which is a free group with free generators a_{1n}, a_{2n}, …, a_{n−1,n}, and

Figure 2. The geometric braid a_{ij}

P_{n−1}. Similarly, P_{n−1} is the semi–direct product of the free group U_{n−1} with free generators a_{1,n−1}, a_{2,n−1}, …, a_{n−2,n−1} and P_{n−2}, and so on. Therefore, P_{n} is decomposable (see [17]) into the following semi–direct product

P_{n} = U_{n} ⋊ (U_{n−1} ⋊ (… ⋊ (U_{3} ⋊ U_{2})) …), U_{i} ≃ F_{i−1}, i = 2, 3, …, n.

The group B_{n} has a faithful representation as a group of automorphisms of Aut(F_{n}) of the free group F_{n} = ⟨x_{1}, x_{2}, …, x_{n}⟩. In this case the generator σ_{i}, i = 1, 2, …, n − 1, defines the automorphism

σ_{i} : \{ \begin{matrix}x_{i} ⟼ x_{i} x_{i+1} x_{i}^{−1}, \\ x_{i+1} ⟼ x_{i}, \\ x_{l} ⟼ x_{l}, & l ≠ i, i + 1. \end{matrix}

By theorem of Artin [4, Theorem 1.9], an automorphism β from Aut(F_{n}) lies in B_{n} if and only if β satisfies the following conditions:

1) β(x_{i}) = a_{i}^{−1} x_{π(i)} a_{i}, 1 ≤ i ≤ n,
2) β(x_{1}x_{2} … x_{n}) = x_{1}x_{2} … x_{n},

where π is a permutation from S_{n} and a_{i} ∈ F_{n}.

An automorphism is called a conjugating automorphism (or a permutation– conjugating automorphism according to the terminology from [10]) if it satisfies to condition 1). The group of conjugating automorphisms C_{n} is generated by σ_{i} and automorphisms α_{i}, i = 1, 2, …, n − 1, where

α_{i} : \{ \begin{matrix}x_{i} ⟼ x_{i+1}, \\ x_{i+1} ⟼ x_{i}, \\ x_{l} ⟼ x_{l}, & l ≠ i, i + 1. \end{matrix}

It is not hard to see that the automorphisms α_{i} generate the symmetric group S_{n} and, hence, satisfy the following relations

(3)  α_{i} α_{i+1} α_{i} = α_{i+1} α_{i} α_{i+1}, i = 1, 2, …, n − 2,
(4)  α_{i} α_{j} = α_{j} α_{i}, |i − j| ≥ 2,
(5)  α_{i}^{2} = 1, i = 1, 2, …, n − 1.

The group C_{n} is defined by relations (1)–(2) of B_{n}, relations (3)–(5) of S_{n}, and the mixed relations (see [10, 19])

(6)  α_{i} σ_{j} = σ_{j} α_{i}, |i − j| ≥ 2,
(7)  σ_{i} α_{i+1} α_{i} = α_{i+1} α_{i} σ_{i+1}, i = 1, 2, …, n − 2,
(8)  σ_{i+1} σ_{i} α_{i+1} = α_{i} σ_{i+1} σ_{i}, i = 1, 2, …, n − 2.

If we consider the group generated by automorphisms ε_{ij}, 1 ≤ i ≠ j ≤ n, where

ε_{ij} : \{ \begin{matrix}x_{i} ⟼ x_{j}^{−1} x_{i} x_{j}, & i ≠ j, \\ x_{l} ⟼ x_{l}, & l ≠ i, \end{matrix}

then we get the group of basis–conjugating automorphisms Cb_{n}. The elements of Cb_{n} satisfy condition 1) for the identical permutation π, i. e., map each generator x_{i} to the conjugating element. J. McCool [18] proved that Cb_{n} is defined by the relations (from here different letters stand for different indices)

(9)  ε_{ij} ε_{kl} = ε_{kl} ε_{ij},
(10)  ε_{ij} ε_{kj} = ε_{kj} ε_{ij},
(11)  (ε_{ij} ε_{kj}) ε_{ik} = ε_{ik} (ε_{ij} ε_{kj}).

The group C_{n} is representable as the semi–direct product: C_{n} = Cb_{n}⋊S_{n}, where S_{n} is generated by the automorphisms α_{1}, α_{2}, …, α_{n−1}. The following equalities are true (see [19]):

ε_{i,i+1} = α_{i} σ_{i}^{−1}, ε_{i+1,i} = σ_{i}^{−1} α_{i},
ε_{ij} = α_{j−1} α_{j−2} … α_{i+1} ε_{i,i+1} α_{i+1} … α_{j−2} α_{j−1} i < j,
ε_{ji} = α_{j−1} α_{j−2} … α_{i+1} α_{i} ε_{i,i+1} α_{i} α_{i+1} … α_{j−2} α_{j−1} i < j.

The structure of Cb_{n} was studied in [2, 3]. There it was proved that Cb_{n}, n ≥ 2, is decomposable into the semi–direct product

Cb_{n} = D_{n−1} ⋊ (D_{n−2} ⋊ (… ⋊ (D_{2} ⋊ D_{1})) …),

of subgroups D_{i}, i = 1, 2, …, n−1, generated by ε_{i+1,1}, ε_{i+1,2}, …, ε_{i+1,i}, ε_{1,i+1}, ε_{2,i+1}, …, ε_{i,i+1}. The elements ε_{i+1,1}, ε_{i+1,2}, …, ε_{i+1,i} generate a free group of rank i. The elements ε_{1,i+1}, ε_{2,i+1}, …, ε_{i,i+1} generate a free abelian group of rank i.

The pure braid group P_{n} is contained in Cb_{n} and the generators of P_{n} can be written in the form

a_{i,i+1} = ε_{i,i+1}^{−1} ε_{i+1,i}^{−1}, i = 1, 2, …, n − 1,
a_{ij} = ε_{j−1,i} ε_{j−2,i} … ε_{i+1,i} (ε_{ij}^{−1} ε_{ji}^{−1}) ε_{i+1,i}^{−1} … ε_{j−2,i}^{−1} ε_{j−1,i}^{−1} =

= ε_{j−1,j}^{−1} ε_{j−2,j}^{−1} … ε_{i+1,j}^{−1} (ε_{ij}^{−1} ε_{ji}^{−1}) ε_{i+1,j} … ε_{j−2,j} ε_{j−1,j}, 2 ≤ i+1 < j ≤ n.

1.2. The singular braid monoid. The Baez–Birman monoid [1, 5] or the singular braid monoid SB_{n} is generated (as a monoid) by elements σ_{i}, σ_{i}^{−1}, τ_{i}, i = 1, 2, …, n − 1. The elements σ_{i}, σ_{i}^{−1} generate the braid group B_{n}. The generators τ_{i} satisfy the defining relations

(12)  τ_{i} τ_{j} = τ_{j} τ_{i}, |i − j| ≥ 2,

other relations are mixed:

(13)  τ_{i} σ_{j} = σ_{j} τ_{i}, |i − j| ≥ 2,
(14)  τ_{i} σ_{i} = σ_{i} τ_{i}, i = 1, 2, …, n − 1,
(15)  σ_{i} σ_{i+1} τ_{i} = τ_{i+1} σ_{i} σ_{i+1}, i = 1, 2, …, n − 2,
(16)  σ_{i+1} σ_{i} τ_{i+1} = τ_{i} σ_{i+1} σ_{i}, i = 1, 2, …, n − 2.

In the work [9] it was proved that the singular braid monoid SB_{n} is embedded into the group SG_{n} which is called the singular braid group and has the same defining relations as SB_{n}.

1.3. The virtual braid group and welded braid group. The virtual braid group V B_{n} was introduced in [15]. In [21] a shorter system of defining relations was found, see below. The group V B_{n} is generated by σ_{i}, ρ_{i}, i = 1, 2, …, n − 1 (see Fig. 3).

The elements σ_{i} generate the braid group B_{n} with defining relations (1)–(2) and the elements ρ_{i} generate the symmetric group S_{n} which is defined by the relations

(17)  ρ_{i} ρ_{i+1} ρ_{i} = ρ_{i+1} ρ_{i} ρ_{i+1}, i = 1, 2, …, n − 2,
(18)  ρ_{i} ρ_{j} = ρ_{j} ρ_{i}, |i − j| ≥ 2,
(19)  ρ_{i}^{2} = 1 i = 1, 2, …, n − 1.

Other relations are mixed:

(20)  σ_{i} ρ_{j} = ρ_{j} σ_{i}, |i − j| ≥ 2,

Figure 3. The geometric virtual braid ρ_{i}

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

Note that the last relation is equivalent to the following relation:

ρ_{i+1} ρ_{i} σ_{i+1} = σ_{i} ρ_{i+1} ρ_{i}.

In the work [12] it was proved that the relations

ρ_{i} σ_{i+1} σ_{i} = σ_{i+1} σ_{i} ρ_{i+1}, ρ_{i+1} σ_{i} σ_{i+1} = ρ_{i} σ_{i+1} σ_{i}.

are not fulfilled in V B_{n}.

The welded braid group W B_{n} was introduced in [10]. This group is generated by σ_{i}, α_{i}, i = 1, 2, …, n − 1. The elements σ_{i} generate the braid group B_{n}. The elements α_{i} generate the symmetric group S_{n} and the following mixed relations hold

(22)  α_{i} σ_{j} = σ_{j} α_{i}, |i − j| ≥ 2,
(23)  σ_{i} α_{i+1} α_{i} = α_{i+1} α_{i} σ_{i+1}, i = 1, 2, …, n − 2,
(24)  σ_{i+1} σ_{i} α_{i+1} = α_{i} σ_{i+1} σ_{i}, i = 1, 2, …, n − 2.

In the work [10] it was proved that W B_{n} is isomorphic to the group of conjugating automorphisms C_{n}.

Comparing the defining relations of V B_{n} with the defining relations of W B_{n}, we see that W B_{n} can be obtained from V B_{n} by adding some new relation. Therefore, there exists a homomorphism

φ_{V W} : V B_{n} ⟶ W B_{n},

taking σ_{i} to σ_{i} and ρ_{i} to α_{i} for all i. Hence, W B_{n} is the homomorphic image of V B_{n}.

In [10] it was proved that the following relation (symmetric to (23))

σ_{i+1} α_{i} α_{i+1} = α_{i} α_{i+1} σ_{i},

is true in W B_{n}. But the following relation is not fulfilled

α_{i+1} σ_{i} σ_{i+1} = σ_{i} σ_{i+1} α_{i}.

The linear representations of V B_{n} and W B_{n} by matrices from GL_{n}(ℤ[t, t^{−1}]) which extends the well known Burau representation was constructed in [21]. The linear representation of C_{n} ≃ W B_{n} it was constructed in [3]. This representation continue (with some conditions on parameters) the known Lawrence–Krammer representation.

2. Generators and defining relations of the virtual pure braid group

In this section we introduce a virtual pure braid group and find its generators and defining relations.

Define the map

ν : V B_{n} ⟶ S_{n}

of V B_{n} onto the symmetric group S_{n} by actions on generators

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

where S_{n} is the group generated by ρ_{i}. The kernel ker(ν) of this map is called the virtual pure braid group and denoted by V P_{n}. It is clear that V P_{n} is a normal subgroup of index n! of V B_{n}. Moreover, since V P_{n} ⋂ S_{n} = e and V B_{n} = V P_{n} · S_{n}, then V B_{n} = V P_{n} ⋊ S_{n}, i. e., the virtual pure braid group is the semi–direct product of V P_{n} and S_{n}.

Define the following elements

λ_{i,i+1} = ρ_{i} σ_{i}^{−1}, λ_{i+1,i} = ρ_{i} λ_{i,i+1} ρ_{i} = σ_{i}^{−1} ρ_{i}, i = 1, 2, …, n − 1,

λ_{ij} = ρ_{j−1} ρ_{j−2} … ρ_{i+1} λ_{i,i+1} ρ_{i+1} … ρ_{j−2} ρ_{j−1},

λ_{ji} = ρ_{j−1} ρ_{j−2} … ρ_{i+1} λ_{i+1,i} ρ_{i+1} … ρ_{j−2} ρ_{j−1}, 1 ≤ i < j − 1 ≤ n − 1. Obviously, all these elements belong to V P_{n} and have the following geometric interpretation (Fig. 4, 5)

The next lemma holds

Lemma 1. Let 1 ≤ i < j ≤ n. The following conjugating rules are fulfilled in V B_{n}:

1) for k < i − 1 or i < k < j − 1 or k > j

ρ_{k} λ_{ij} ρ_{k} = λ_{ij}, ρ_{k} λ_{ji} ρ_{k} = λ_{ji};

2) ρ_{i−1} λ_{ij} ρ_{i−1} = λ_{i−1,j}, ρ_{i−1} λ_{ji} ρ_{i−1} = λ_{j,i−1};

3) for i < j − 1 Figure 4. The geometric virtual braid λ_{ij} (1 ≤ i < j ≤ n)

Figure 5. The geometric virtual braid λ_{ji} (1 ≤ i < j ≤ n)

\begin{matrix}ρ_{i} λ_{i,i+1} ρ_{i} = λ_{i+1,i}, & ρ_{i} λ_{ij} ρ_{i} = λ_{i+1,j}, \\ ρ_{i} λ_{i+1,i} ρ_{i} = λ_{i,i+1}, & ρ_{i} λ_{ji} ρ_{i} = λ_{j,i+1}; \end{matrix}

4) for i + 1 < j

ρ_{j−1} λ_{ij} ρ_{j−1} = λ_{i,j−1}, ρ_{j−1} λ_{ji} ρ_{j−1} = λ_{j−1,i};

5) ρ_{j} λ_{ij} ρ_{j} = λ_{i,j+1}, ρ_{j} λ_{ji} ρ_{j} = λ_{j+1,i}.

Proof. We consider only the rules containing λ_{ij} for i < j (the remaining rules can be considered analogously). Recall that

λ_{ij} = ρ_{j−1} ρ_{j−2} … ρ_{i+1} λ_{i,i+1} ρ_{i+1} … ρ_{j−2} ρ_{j−1}.

If k < i − 1 or k > j then ρ_{k} is permutable with ρ_{i}, ρ_{i+1}, …, ρ_{j−1} in view of relation (18) and with σ_{i} in view of relation (20). Hence, ρ_{k} is permutable with λ_{ij}.

10 BARDAKOV

Let i < k < j − 1. Then ρ_{k} λ_{ij} ρ_{k} = ρ_{k} (ρ_{j−1} … ρ_{k+2} ρ_{k+1} ρ_{k} … ρ_{i+1} λ_{i,i+1} ρ_{i+1} … ρ_{k} ρ_{k+1} ρ_{k+2} … ρ_{j−1}) ρ_{k}. Permuting ρ_{k} to λ_{i,i+1} when it is possible, we get ρ_{j−1} … ρ_{k+2} (ρ_{k} ρ_{k+1} ρ_{k}) … ρ_{i+1} λ_{i,i+1} ρ_{i+1} … (ρ_{k} ρ_{k+1} ρ_{k}) ρ_{k+2} … ρ_{j−1}. Using the relation ρ_{k} ρ_{k+1} ρ_{k} = ρ_{k+1} ρ_{k} ρ_{k+1}, rewrite the last formula as follows:

ρ_{j−1} … ρ_{k+2} ρ_{k+1} ρ_{k} (ρ_{k+1} ρ_{k−1} … ρ_{i+1} λ_{i,i+1} ρ_{i+1} … ρ_{k−1} ρ_{k+1})×

×ρ_{k} ρ_{k+1} ρ_{k+2} … ρ_{j−1} = ρ_{j−1} … ρ_{k} (ρ_{k+1} λ_{i,k} ρ_{k+1}) ρ_{k} … ρ_{j−1}.

In view of the case considered earlier, we have

ρ_{k+1} λ_{ik} ρ_{k+1} = λ_{ik}

and, hence,

ρ_{j−1} … ρ_{k} (ρ_{k+1} λ_{ik} ρ_{k+1}) ρ_{k} … ρ_{j−1} = λ_{ij}.

Thus, the first rule from 1) is proven.

2) Consider

ρ_{i−1} λ_{ij} ρ_{i−1} = ρ_{i−1} (ρ_{j−1} ρ_{j−2} … ρ_{i+1} λ_{i,i+1} ρ_{i+1} … ρ_{j−2} ρ_{j−1}) ρ_{i−1}. Using relation (18), let as permute ρ_{i−1} to λ_{i,i+1} when it is possible. We get

(25) ρ_{i−1} λ_{ij} ρ_{i−1} = ρ_{j−1} … ρ_{i+2} ρ_{i+1}(ρ_{i−1} λ_{i,i+1} ρ_{i−1}) ρ_{i+1}ρ_{i+2} … ρ_{j−2}.

The expression in the brackets can be rewritten in the following form

ρ_{i−1} λ_{i,i+1} ρ_{i−1} = ρ_{i−1} ρ_{i} σ_{i}^{−1} ρ_{i−1} = ρ_{i−1} ρ_{i} σ_{i}^{−1} ρ_{i−1}ρ_{i} ρ_{i}. Using the relation σ_{i}^{−1} ρ_{i−1} ρ_{i} = ρ_{i−1} ρ_{i} σ_{i−1}^{−1} (it follows from (21)) and (18), (19), we obtain

ρ_{i−1} ρ_{i} (σ_{i}^{−1} ρ_{i−1} ρ_{i}) ρ_{i} = ρ_{i−1} (ρ_{i} ρ_{i−1} ρ_{i}) σ_{i−1}^{−1} ρ_{i} =

= (ρ_{i−1} ρ_{i−1}) ρ_{i} ρ_{i−1} σ_{i−1}^{−1} ρ_{i} = ρ_{i} λ_{i−1,i} ρ_{i}.

Then from (25) we obtain

ρ_{i−1} λ_{ij} ρ_{i−1} = λ_{i−1,j}.

Therefore, the desired relations are proven.

3) The first formula follows from the definitions of λ_{i,i+1} and λ_{i+1,i}. Let us consider

ρ_{i} λ_{ij} ρ_{i} = ρ_{i} (ρ_{j−1} ρ_{j−2} … ρ_{i+1} λ_{i,i+1} ρ_{i+1} … ρ_{j−2} ρ_{j−1}) ρ_{i}. Permuting ρ_{i} to λ_{i,i+1} when it is possible, we obtain

ρ_{i} λ_{ij} ρ_{i} = ρ_{j−1} … ρ_{i+2} (ρ_{i} ρ_{i+1} λ_{i,i+1} ρ_{i+1} ρ_{i}) ρ_{i+2} … ρ_{j−1}.

Rewrite the expression in the brackets as follows

ρ_{i} ρ_{i+1} λ_{i,i+1} ρ_{i+1} ρ_{i} = ρ_{i} ρ_{i+1} ρ_{i} (σ_{i}^{−1} ρ_{i+1} ρ_{i}) = ρ_{i} ρ_{i+1} (ρ_{i} ρ_{i+1} ρ_{i}) σ_{i+1}^{−1} =

= ρ_{i} ρ_{i+1} ρ_{i+1} ρ_{i} ρ_{i+1} σ_{i+1}^{−1} = ρ_{i+1} σ_{i+1}^{−1}.

Hence,

ρ_{i} λ_{ij} ρ_{i} = ρ_{j−1} … ρ_{i+2} (ρ_{i+1} σ_{i+1}^{−1}) ρ_{i+2} … ρ_{j−1} = λ_{i+1,j}.

Therefore, the desired relations are proven.

4) follows from the relation ρ_{j−1}^{2} = e and the definition of λ_{ij}.

5) is an immediate consequence of the definition of λ_{ij}. □

Corollary 1. The group S_{n} acts by conjugation on the set {λ_{kl} |1 ≤ k ≠ l ≤ n}. This action is transitive.

In view of Lemma 1, the subgroup ⟨λ_{kl} | 1 ≤ k ≠ l ≤ n⟩ of V P_{n} is normal in V B_{n}. Let us prove that this group coincides with V P_{n} and let us find its generators and defining relations. For this purpose we use the Reidemeister–Schreier method (see, for example, [16, Ch. 2.2]).

Let m_{kl} = ρ_{k−1} ρ_{k−2} … ρ_{l} for l < k and m_{kl} = 1 in other cases. Then the set

Λ_{n} = {∏_{k=2}^{n} m_{k,j_{k}}|1 ≤ j_{k} ≤ k}

is a Schreier set of coset representatives of V P_{n} in V B_{n}.

Theorem 1. The group V P_{n} admits a presentation with the generators λ_{kl}, 1 ≤ k ≠ l ≤ n, and the defining relations:

(26)  λ_{ij} λ_{kl} = λ_{kl} λ_{ij},
(27)  λ_{ki} (λ_{kj} λ_{ij}) = (λ_{ij} λ_{kj}) λ_{ki},

where distinct letters stand for distinct indices.

Proof. Define the map ^{−} : V B_{n} ⟶ Λ_{n} which takes an element w ∈ V B_{n} into the representative \overline{w} from Λ_{n}. In this case the element w\overline{w}^{−1} belongs to V P_{n}. By Theorem 2.7 from [16] the group V P_{n} is generated by

s_{λ,a} = λa · (\overline{λa})^{−1},

where λ runs over the set Λ_{n} and a runs over the set of generators of V B_{n}.

It is easy to establish that s_{λ,ρ_{i}} = e for all representatives λ and generators ρ_{i}. Consider the generators

s_{λ,σ_{i}} = λσ_{i} · (\overline{λρ_{i}})^{−1}.

For λ = e we get s_{e,σ_{i}} = σ_{i}ρ_{i} = λ_{i,i+1}^{−1}. Note that λρ_{i} is equal to \overline{λρ_{i}} in S_{n}. Therefore,

s_{λ,σ_{i}} = λ(σ_{i}ρ_{i})λ^{−1}.

From Lemma 1 it follows that each generator s_{λ,σ_{i}} is equal to some λ_{kl}, 1 ≤ k ≠ l ≤ n. By Corollary 1, the inverse statement is also true, i. e., each element λ_{kl} is equal to some generator s_{λ,σ_{i}}. The first part of the theorem is proven.

To find defining relations of V P_{n} we define a rewriting process τ. It allows us to rewrite a word which is written in the generators of V B_{n} and presents an element in V P_{n} as a word in the generators of V P_{n}. Let us associate to the reduced word

u = a_{1}^{ε_{1}} a_{2}^{ε_{2}} … a_{ν}^{ε_{ν}}, ε_{l} = ±1, a_{l} ∈ {σ_{1}, σ_{2}, …, σ_{n−1}, ρ_{1}, ρ_{2}, …, ρ_{n−1}},

the word

τ(u) = s_{k_{1},a_{1}}^{ε_{1}} s_{k_{2},a_{2}}^{ε_{2}} … s_{k_{ν},a_{ν}}^{ε_{ν}}

in the generators of V P_{n}, where k_{j} is a representative of the (j − 1)th initial segment of the word u if ε_{j} = 1 and k_{j} is a representative of the jth initial segment of the word u if ε_{j} = −1.

By [16, Theorem 2.9], the group V P_{n} is defined by relations

r_{μ,λ} = τ(λ r_{μ} λ^{−1}), λ ∈ Λ_{n},

where r_{μ} is the defining relation of V B_{n}.

Denote by

r_{1} = σ_{i} σ_{i+1} σ_{i} σ_{i+1}^{−1} σ_{i}^{−1} σ_{i+1}^{−1}

the first relation of V B_{n}. Then

r_{1,e} = τ(r_{1}) = s_{e,σ_{i}} s_{\overline{σ_{i}},σ_{i+1}} s_{\overline{σ_{i}σ_{i+1}},σ_{i}} s_{\overline{σ_{i}σ_{i+1}σ_{i}σ_{i+1}^{−1}},σ_{i+1}}^{−1} s_{\overline{σ_{i}σ_{i+1}σ_{i}σ_{i+1}^{−1}σ_{i}^{−1}},σ_{i}}^{−1} s_{\overline{r_{1}},σ_{i+1}}^{−1} =

= λ_{i,i+1}^{−1} (ρ_{i} λ_{i+1,i+2}^{−1} ρ_{i}) (ρ_{i} ρ_{i+1} λ_{i,i+1}^{−1} ρ_{i+1} ρ_{i})×
×(ρ_{i+1} ρ_{i} λ_{i+1,i+2} ρ_{i} ρ_{i+1}) (ρ_{i+1} λ_{i,i+1} ρ_{i+1}) λ_{i+1,i+2}.

Using the conjugating rules from Lemma 1, we get

r_{1,e} = λ_{i,i+1}^{−1} λ_{i,i+2}^{−1} λ_{i+1,i+2}^{−1} λ_{i,i+1} λ_{i,i+2} λ_{i+1,i+2}.

Therefore, the following relation

λ_{i,i+1} (λ_{i,i+2} λ_{i+1,i+2}) = (λ_{i+1,i+2} λ_{i,i+2}) λ_{i,i+1}

is fulfilled in V P_{n}. The Remaining relations r_{1,λ}, λ ∈ Λ_{n}, can be obtained from this relation using conjugation by λ^{−1}. By the formulas from Lemma 1, we obtain relations (27).

Let us consider the next relation of V B_{n}:

r_{2} = σ_{i} σ_{j} σ_{i}^{−1} σ_{j}^{−1}, |i − j| ≥ 2.

For it we have

r_{2,e} = τ(r_{2}) = s_{e,σ_{i}} s_{\overline{σ_{i}},σ_{j}} s_{\overline{σ_{i}σ_{j}σ_{i}^{−1}},σ_{i}}^{−1} s_{\overline{r_{2}},σ_{j}}^{−1} =
= λ_{i,i+1}^{−1} λ_{j,j+1}^{−1} λ_{i,i+1} λ_{j,j+1}.

Hence, the relation

λ_{i,i+1} λ_{j,j+1} = λ_{j,j+1} λ_{i,i+1}, |i − j| ≥ 2

holds in V P_{n}. Conjugating this relation by all representatives from Λ_{n}, we obtain relations (26).

Let us prove that only trivial relations follow from all other relations of V B_{n}. It is evident for relations (17)–(19) defining the group S_{n} because s_{λ,ρ_{i}} = e for all λ ∈ Λ_{n} and ρ_{i}.

Consider the mixed relation (21) (relation (20) can be considered similarly):

r_{3} = σ_{i+1} ρ_{i} ρ_{i+1} σ_{i}^{−1} ρ_{i+1} ρ_{i}.

Using the rewriting process, we get

r_{3,e} = τ(r_{3}) = s_{e,σ_{i+1}} s_{\overline{σ_{i+1}ρ_{i}ρ_{i+1}σ_{i}^{−1}},σ_{i}}^{−1} =
= λ_{i+1,i+2}^{−1} (ρ_{i} ρ_{i+1} λ_{i,i+1} ρ_{i+1} ρ_{i}) = λ_{i+1,i+2}^{−1} λ_{i+1,i+2} = e.

Therefore, V P_{n} is defined by relations (26) –(27). □

3. The structure of the virtual braid group

From the definition of V P_{n} and Lemma 1 it follows that V B_{n} = V P_{n} ⋊ S_{n}, i. e., V B_{n} is the splittable extension of the group V P_{n} by S_{n}. Consequently, we have to study the structure of the virtual pure braid group V P_{n}. Let us define the subgroups

V_{i} = ⟨λ_{1,i+1}, λ_{2,i+1}, …, λ_{i,i+1}; λ_{i+1,1}, λ_{i+1,2}, …, λ_{i+1,i}⟩, i = 1, 2, …, n−1,

of V P_{n}. Each V_{i} is a subgroup of V P_{i+1}. Let V_{i}^{∗} be the normal closure of V_{i} in V P_{i+1}. The following theorem is the main result of this section.

Theorem 2. The group V P_{n}, n ≥ 2, is representable as the semi–direct product

V P_{n} = V_{n−1}^{∗} ⋊ V P_{n−1} = V_{n−1}^{∗} ⋊ (V_{n−2}^{∗} ⋊ (… ⋊ (V_{2}^{∗} ⋊ V_{1}^{∗})) …),

where V_{1}^{∗} is a free group of rank 2 and V_{i}^{∗}, i = 2, 3, …, n − 1, are free infinitely generated subgroups.

Let us prove the theorem by induction on n. For n = 2, we have

V P_{2} = V_{1} = V_{1}^{∗}

and, by Theorem 1, the group V_{1} is free generated by λ_{12} and λ_{21}. To make the general case more clear consider the case n = 3.

3.1. The structure of V P_{3}. By Theorem 1, the group V P_{3} is generated by subgroups V_{1}, V_{2} and defined by the relations

λ_{12} (λ_{13} λ_{23}) = (λ_{23} λ_{13}) λ_{12}, λ_{21} (λ_{23} λ_{13}) = (λ_{13} λ_{23}) λ_{21},
λ_{13} (λ_{12} λ_{32}) = (λ_{32} λ_{12}) λ_{13}, λ_{31} (λ_{32} λ_{12}) = (λ_{12} λ_{32}) λ_{31},
λ_{23} (λ_{21} λ_{31}) = (λ_{31} λ_{21}) λ_{23}, λ_{32} (λ_{31} λ_{21}) = (λ_{21} λ_{31}) λ_{32}.

From these relations we obtain the next lemma.

Lemma 2. In V P_{3} the following equalities hold: 1)

\begin{matrix}λ_{13}^{λ_{12}} = λ_{32}^{λ_{12}} λ_{13} λ_{32}^{−1}, & λ_{31}^{λ_{12}} = λ_{32} λ_{31} λ_{32}^{−λ_{12}}, & λ_{23}^{λ_{12}} = λ_{13} λ_{23} λ_{32} λ_{13}^{−1} λ_{32}^{−λ_{12}}, \\ λ_{13}^{λ_{12}^{−1}} = λ_{32}^{−1} λ_{13} λ_{32}^{λ_{12}^{−1}}, & λ_{31}^{λ_{12}^{−1}} = λ_{32}^{−λ_{12}^{−1}} λ_{31}λ_{32}, & λ_{23}^{λ_{12}^{−1}} = λ_{32}^{−λ_{12}^{−1}} λ_{13}^{−1} λ_{32} λ_{23} λ_{13}, \end{matrix}

2)

\begin{matrix}λ_{23}^{λ_{21}} = λ_{31}^{λ_{21}} λ_{23} λ_{31}^{−1}, & λ_{32}^{λ_{21}} = λ_{31} λ_{32} λ_{31}^{−λ_{21}}, & λ_{13}^{λ_{21}} = λ_{23} λ_{13} λ_{31} λ_{23}^{−1} λ_{31}^{−λ_{21}}, \\ λ_{23}^{λ_{21}^{−1}} = λ_{31}^{−1} λ_{23} λ_{31}^{λ_{21}^{−1}}, & λ_{32}^{λ_{21}^{−1}} = λ_{31}^{−λ_{21}^{−1}} λ_{32} λ_{31}, & λ_{13}^{λ_{21}^{−1}} = λ_{31}^{−λ_{21}^{−1}} λ_{23}^{−1} λ_{31} λ_{13} λ_{23}, \end{matrix}

where a^{b} stand for b^{−1}ab.

Proof. The first and second relations from 1) immediately follow from the third and forth relations of V P_{3} (see the relations before the lemma). Similarly, the first and second relations from 2) immediately follow from the fifth and sixth relations of V P_{3}.

Further, from the first and second relations of V P_{3} we obtain

λ_{23}^{λ_{1,2}} = λ_{13} λ_{23} λ_{13}^{−λ_{12}}, λ_{13}^{λ_{21}} = λ_{23} λ_{13} λ_{23}^{−λ_{21}}.

Using the proved formulas for λ_{13}^{λ_{12}} and λ_{23}^{λ_{21}}, we get the third formulas from 1) and 2) respectively.

The formulas for conjugation by λ_{12}^{−1} and λ_{21}^{−1} can be obtained analogously. □

Note that there exists an epimorphism

φ_{3} : V P_{3} ⟶ V P_{2},

which takes the generators of V_{2} = ⟨λ_{13}, λ_{23}, λ_{31}, λ_{32}⟩ into the unit and fixes the generators of V_{1} = ⟨λ_{12}, λ_{21}⟩. The kernel of this epimorphism is the normal closure of V_{2} in V P_{3}, i. e., ker(φ_{3}) = V_{2}^{∗}.

Let u be the empty word or a reduced word beginning with a non-zero power of λ_{12} and representing an element from V_{1}. Let λ_{32}(u) = λ_{32}^{u} = u^{−1} λ_{32} u. We call this element the reduced power of the generator λ_{32} with the power u. Analogously, if v is the empty word or a reduced word beginning with a non-zero power of λ_{21} and representing an element from V_{1}, then we put λ_{31}(v) = λ_{13}^{v} and call this element the reduced power of generator λ_{31} with the power v.

Lemma 3. The group V_{2}^{∗} is a free group with generators λ_{13}, λ_{23} and all reduced powers of λ_{31} and λ_{32}.

Proof. To prove the lemma we can use the Reidemeister–Shreier method, but it is simpler to use the definitions of normal closure and semi-direct product. Evidently, the group V_{2}^{∗} is generated by the elements

λ_{13}^{w}, λ_{23}^{w}, λ_{31}^{w}, λ_{32}^{w}, w ∈ V_{1}.

In view of Lemma 2, it is sufficient to take from these elements only λ_{13}, λ_{23} and all reduced powers of the generators λ_{31} and λ_{32}.

The freedom of V_{2}^{∗} follows from the representation of V P_{3} as the semi–direct product. Indeed, since V_{1} ⋂ V_{2}^{∗} = e, V_{1}V_{2}^{∗} = V P_{3}, then V P_{3} = V_{2}^{∗}⋊V_{1}. In this case the defining relations of V P_{3} are equivalent to the conjugating rules from Lemma 2. Therefore, all relations define the action of the group V_{1} on the group V_{2}^{∗}. Since there are no other relations, this means that V_{1} and V_{2}^{∗} are free groups. □

As a consequence of this Lemma, we obtain the normal form of words in V P_{3}. Any element w from V P_{3} can be written in the form w = w_{1}w_{2}, where w_{1} is a reduced word over the alphabet {λ_{12}^{±1}, λ_{21}^{±1}} and w_{2} is a reduced word over the alphabet {λ_{13}^{±1}, λ_{23}^{±1}, λ_{31}(u)^{±1}, λ_{32}(v)^{±1}}, where λ_{31}(u), λ_{32}(v) are reduced powers of the generators λ_{31} and λ_{32} respectively.

3.2. The proof of Theorem 2. Now, we introduce the following notation. By λ_{ij}^{∗} denote any λ_{ij} or λ_{ji} from V P_{n}.

Lemma 4. For every n ≥ 2 there exists a homomorphism

φ : V P_{n} ⟶ V P_{n−1}

which takes the generators λ_{ij}^{∗}, i = 1, 2, …, n − 1, to the unit and fixes other generators.

Proof. It is sufficient to prove that all defining relations go to the defining relations by a such defined map. For the defining relations of V P_{n−1} it is evident. If the relation of commutativity (see relation (26)) contains some generator of V_{n−1} then by acting with φ_{n} it turns to the trivial relation. Let us consider the left hand side of relation (27). We see that it contains every index two times. Hence, if this part includes some generator of V_{n−1} (i. e., one of the indices is equal to n) then some other generator contains the index n. Therefore, there are two generators of V_{n−1} in the left hand side of the relation. Since the right hand side contains all generators from the left hand side, then by acting with φ_{n} this relation turns to the trivial relation. □

Lemma 5. The following formulas are fulfilled in the group V P_{n}:

1) λ_{kl}^{λ_{ij}^{ε}} = λ_{kl}, max{i, j} < max{k, l}, ε = ±1;

2) λ_{ik}^{λ_{ij}} = λ_{kj}^{λ_{ij}}λ_{ik}λ_{kj}^{−1}, λ_{ik}^{λ_{ij}^{−1}} = λ_{kj}^{−1}λ_{ik}λ_{kj}^{λ_{ij}^{−1}}, i < j < k or j < i < k;

3) λ_{ki}^{λ_{ij}} = λ_{kj}λ_{ki}λ_{kj}^{−λ_{ij}}, λ_{ki}^{λ_{ij}^{−1}} = λ_{kj}^{−λ_{ij}^{−1}}λ_{ki}λ_{kj}, i < j < k or j < i < k;

4) λ_{jk}^{λ_{ij}} = λ_{ik}λ_{jk}λ_{kj}λ_{ik}^{−1}λ_{kj}^{−λ_{ij}}, λ_{jk}^{λ_{ij}^{−1}} = λ_{jk}^{−λ_{ik}^{−1}}λ_{ij}^{−1}λ_{jk}λ_{kj}λ_{ij}, i < j < k or j < i < k, where, as usual, different letters stand for different indices.

Proof. The formula 1) immediately follows from the first relation of Theorem 1.

Consider relation (27) from Theorem 1:

λ_{ki} (λ_{kj} λ_{ij}) = (λ_{ij} λ_{kj}) λ_{ki}.

Note that the indices of generators are connected by one of the inequalities:

a) k < j < i, b) j < k < i, c) i < j < k,
d) j < i < k, e) k < i < j, f) i < k < j.

If the indices are connected by inequality a) or b) then from (27) we obtain

λ_{ki}^{λ_{kj}} = λ_{ij}^{λ_{kj}} λ_{ki} λ_{ij}^{−1},

and it is the first formula from 2).

If the indices in relation (27) are connected by inequality c) or d) we obtain

λ_{ki}^{λ_{ij}} = λ_{kj} λ_{ki} λ_{kj}^{−λ_{ij}},

and it is the first formula from 3).

If indices in relation (27) are connected by inequality e) or f) then

λ_{ij}^{λ_{ki}} = λ_{kj} λ_{ij} λ_{kj}^{−λ_{ki}}.

Using the formula from 2), we obtain

λ_{ij}^{λ_{ki}} = λ_{kj} λ_{ij} λ_{ji} λ_{kj}^{−1} λ_{ji}^{−λ_{ki}},

and it is the first formula from 4).

The formulas of conjugations by elements λ_{ij}^{−1} can be established similarly.

□

Assume that the theorem is proven for the group V P_{n−1}. Hence, any element w ∈ V P_{n−1} can be written in the form

w = w_{1} w_{2} … w_{n−2}, w_{i} ∈ V_{i}^{∗},

where each word w_{i} is a reduced word over the alphabet consisting of generators λ_{ki}^{±1}, 1 ≤ k ≤ i − 1, and reduced powers of generators λ_{ki}, 1 ≤ k ≤ i − 1, and their inverse. Let us define reduced powers of generators in the group V_{n−1}^{∗}. We say that the element λ_{nk}(w) = λ_{nk}^{w} is the reduced power of the generator λ_{nk} if w is the empty word or a word written in the normal form and begining with a reduced power of some generator λ_{lk} or its inverse.

The statement about decomposition as the semi–direct product V P_{n} = V_{n}^{∗} ⋊ V P_{n−1} is quite evident. It remains to find generators of V_{n}^{∗} and prove its freedom.

Lemma 6. The group V_{n−1}^{∗} is a free group. It is generated by λ_{1n}, λ_{2n}, …, λ_{n−1,n} and all reduced powers of the generators λ_{n1}, λ_{n2}, …, λ_{n,n−1}.

Proof. The proof is similar to that of Lemma 3. From Lemma 5 it follows that this set is the set of generators of V_{n−1}^{∗}. Further, since the set of defining relations of V P_{n} is equivalent to the set of conjugating formulas defining the action of V P_{n−1} on V_{n−1}^{∗}, only trivial relations are fulfilled in V_{n−1}^{∗}. □

Theorem 2 follows from these results.

As a consequence of this theorem we obtain the normal form of words in V B_{n}.

Corollary 2. Every element from V B_{n} can be written uniquely in the form

w = w_{1} w_{2} … w_{n−1} λ, λ ∈ Λ_{n}, w_{i} ∈ V_{i}^{∗},

where w_{i} is a reduced word in generators, reduced powers of generators and their inverse.

The homomorphism defined above of the virtual braid group onto the welded braid group agrees with the decomposition from Theorem 2 and with the decomposition of C_{n} ≃ W B_{n} described in the first section.

Corollary 3. The homomorphism φ_{V W} : V B_{n} ⟶ W B_{n} agrees with the decomposition of these groups, i. e., it maps the group V P_{n} onto Cb_{n} ≃ W P_{n} and the factors V_{i}^{∗} onto the factors D_{i}, i = 1, 2, …, n − 1.

4. The universal braid group

Let us define the universal braid group UB_{n} as the group with generators σ_{1}, σ_{2}, …, σ_{n−1}, c_{1}, c_{2}, …, c_{n−1}, defining relations (1)–(2), the relations:

c_{i} c_{j} = c_{j} c_{i}, |i − j| ≥ 2,

and the mixed relations:

c_{i} σ_{j} = σ_{j} c_{i} |i − j| ≥ 2.

Recall (see [6]) that Artin’s groups of the type I is called the group A_{I} with generators a_{i}, i ∈ I, and the defining relations

a_{i} a_{j} a_{i} … = a_{j} a_{i} a_{j} …, i, j ∈ I,

where words from the left and right hand sides consist of m_{ij} alternating letters a_{i} and a_{j}.

Proposition 1. 1) The group UB_{n} has as a subgroup the braid group B_{n}.

2) There exist surjective homomorphisms

φ_{US} : UB_{n} ⟶ SG_{n}, φ_{UV} : UB_{n} ⟶ V B_{n}, φ_{UB} : UB_{n} ⟶ B_{n}.

3) The group UB_{n} is Artin’s group.

Proof. 1) Evidently, there exists a homomorphism B_{n} ⟶ UB_{n}. On the other hand, assuming ψ(σ_{i}) = σ_{i}, ψ(c_{i}) = e, i = 1, 2, …, n − 1, we obtain the retraction ψ of UB_{n} onto B_{n}. Therefore, the subgroup ⟨σ_{1}, σ_{2}, …, σ_{n−1}⟩ of UB_{n} is isomorphic to the braid group B_{n}.

2) Let us define the map φ_{US} as follows

φ_{US}(σ_{i}) = σ_{i}, φ_{US}(c_{i}) = τ_{i}, i = 1, 2, …, n − 1.

Comparing the defining relations of UB_{n} and SG_{n}, we see that this map is a homomorphism. Analogously, we can show that the map

σ_{i} ⟼ σ_{i}, c_{i} ⟼ ρ_{i},

is extendable to the homomorphism φ_{UV} and the map

σ_{i} ⟼ σ_{i}, c_{i} ⟼ e,

is extendable to the homomorphism φ_{UB}.

3) immediately follows from the defining relations of UB_{n} and the definition of Artin’s group. □

It should be noted that none of the groups SG_{n}, V B_{n}, W B_{n} (in the natural presentations) is not Artin’s group.

The following questions naturally arise in the context of the results obtained above.

Problems. 1) Solve the word and conjugacy problems in UB_{n}, n > 2.

2) Is it possible to give some geometric interpretation for elements of UB_{n} similar to the geometric interpretation for elements of the braid groups B_{n}, SG_{n}, V B_{n}, UB_{n}?

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Sobolev Institute of Mathematics, Novosibirsk 630090, Russia

  E-mail address: bardakov@math.nsc.ru