Representations of flat virtual braids by automorphisms of free group

Representations of braid group $B_n$ on $n \geq 2$ strands by automorphisms of a free group of rank $n$ go back to Artin (1925). In 1991 Kauffman introduced a theory of virtual braids and virtual knots and links. The virtual braid group $VB_n$ on $n \geq 2$ strands is an extension of the classical braid group $B_n$ by the symmetric group $S_n$. In this paper we consider flat virtual braid groups $FVB_n$ on $n\geq 2$ strands and construct a family of representations of $FVB_n$ by automorphisms of free groups of rank $2n$. It has been established that these representations do not preserve the forbidden relations between classical and virtual generators. We investigated some algebraic properties of the constructed representations. In particular, we established conditions of faithfulness in case $n=2$ and proved that the kernel contains a free group of rank two for $n\ge3$.


Introduction
The foundations of the braid groups theory were laid down in the works of E. Artin in the 1920s.In [1] he defined the braid group B n on n ≥ 2 strands as a group with generators σ 1 , . . ., σ n−1 and defining relations: A set {σ 1 , . . ., σ n−1 } is called the standard generators, or the Artin generators of the braid group B n .The generator σ i ∈ B n and its inverse σ −1 i are presented geometrically in Figure 1.
Generator σ i ∈ B n and its inverse σ −1 i .
There is very nice relation between braid groups and knot theory based on Alexender's theorem and Markov's theorem [2].Invariants, arising from representations of braid groups, play an important role in classical knot theory and its generalizations.
Artin discovered faithful representation ϕ n : B n → Aut(F n ), where F n = x 1 , . . ., x nis the free group of rank n.Homomorphism ϕ n maps generator σ i ∈ B n to the following automorphism ϕ n (σ i ): Note, that for each i, Moreover, it is shown by Artin [1,3] that an automorphism g ∈ Aut(F n ) is equal to ϕ n (β) for some β ∈ B n if and only if the following two conditions are satisfied: (i) f (x i ) is conjugate of some x j for i = 1, . . ., n − 1; and (ii) In what follows, if any automorphism acts on a generator identically, we will not write this action.We write the composition of automorphisms in the order of their application from left to right, namely, ϕψ(f ) = ψ(ϕ(f )).
S. Kamada [6] established that the following Alexander's theorem for virtual braids holds: If K is a virtual link, then for some n there exists a virtual braid β ∈ V B n such that K is the closure of β.
It is known as shown in [7] that relations do not hold in the V B n group.These relations ( 8) -( 9) are called forbidden relations, see Figures 4 and 5.The group W B n is obtained from V B n by adding the relation (8) and is called welded braid group [8].The same group W B n is obtained by adding the relation (9) to the group V B n .Adding both relations (8) and (9) to V B n leads to unknotting transformations for virtual knots and links [7,9,6].Other unknotting operations for links, virtual links and welded links are given, for example, in [10,11,12].Note that the representations V B n → Aut(G n ) were constructed, for example, for groups G n of the following form: [15].For structural properties and other representations of the virtual braid groups see [16,17].
In [18] the group of flat virtual braids F V B n on n strands was introduced as a result of adding the relations (10) to the group V B n : We summarize the above discussions in the following definition.
Generator σ i ∈ F V B n is presented geometrically in Figure 6 and generator ρ ∈ F V B n is presented geometrically in Figure 2.
In [19] the following problem was formulated: Does it exist a representation of the F V B n group by automorphisms of some group for which the forbidden relations would not hold?In [20] such representation η n : F V B n → Aut(F 2n ) was constructed, here F 2n = x 1 , . . ., x n , y 1 , . . ., y n is a free group of rank 2n.The homomorphism η n maps generators σ i , ρ i ∈ F V B n , where i = 1, . . ., n − 1, to the following automorphisms: It was shown in [20] that the representation In this paper we construct a family of representations of the F V B n group by automorphisms of the free group F 2n = x 1 , . . ., x n , y 1 , . . ., y n , which generalize the representation (11).Namely, we consider a family of homomorphisms Θ n : F V B n → Aut(F 2n ), which are given by mapping generators σ i , ρ i ∈ F V B n , where i = 1, . . ., n − 1, to the following automorphisms: (12) Θ n (σ i ) : where the elements a i (y i , y i+1 ), b i (y i , y i+1 ), c i (y i , y i+1 ) and d i (y i , y i+1 ) are words in a free group of rank two with generators {y i , y i+1 } for each i = 1, . . ., n − 1.Thus, the homomorphisms Θ n depend only on the choice of the words a i , b i , c i , d i , which define the locally nontrivial action of the automorphism corresponding to the generator of the group F V B n , and in this sense the homomorphisms Θ n are local homomorphisms.
The article has the following structure.In Theorem 1 we establish for which a i , b i , c i and d i there exists a local homomorphism Θ n of the group F V B n into the automorphism group of the free group F 2n .In Section 3 we obtain results about the structure of the kernel of the homomorphism Θ n , in particular, in Theorem 3 we describe the kernel of this homomorphism for n = 2.In Theorem 4 it will be established that for n ≥ 3 the kernel of the homomorphism Θ n contains a free group of rank 2. We note it was shown earlier in [20] that for n ≥ 3 the kernel of the homomorphism η n , which is a special case of Θ n , contains an infinite cyclic group.

Existence of local representations
Let F 2n be a free group of rank 2n with free generators x 1 , . . ., x n , y 1 , . . ., y n .Theorem 1.Let n ≥ 2 and a i (y i , y i+1 ), b i (y i , y i+1 ), c i (y i , y i+1 ), d i (y i , y i+1 ) are words in a free group of rank two with generators {y i , y i+1 }, where 1 ≤ i ≤ n − 1. Define the map Θ n : F V B n → Aut(F 2n ) by mapping σ i and ρ i to automorphisms: (13) Θ n (σ i ) : Then the following properties hold.
(1) The map Θ n is homomorphism iff where m i ∈ Z for 1 ≤ i ≤ n − 1, and with n ≥ 3, where a 1 = w(y 1 , y 2 ) for some word w(A, B) ∈ The map Θ n does not preserve the forbidden relations.
Proof.(1) Let us verify that the relations ( 1) -( 7) and the relation (10) are preserved under the map Θ n .Denote The relations (1), ( 4) and ( 6) are preserved because s i acts non-trivially only on x i and x i+1 , while r i acts non-trivially only on x i , x i+1 , y i and y i+1 .Since s 2 i : i : and relation (3) is preserved iff Consider the actions of automorphisms r i r i+1 s i and s i+1 r i r i+1 .We have ).Thus, to fulfill the relation (7), it is necessary and sufficient that a i+1 (y i , y i+1 )c i (y i+2 , y i ) = c i+1 (y i+2 , y i+1 )a i (y i , y i+1 ), (15) holds for all 1 ≤ i ≤ n − 2.
A similar consideration of the relations (5) leads to the equalities for all 1 ≤ i ≤ n − 2. This is only possible if c i (y i , y i+1 ) = y m i i+1 for some m i ∈ Z with 1 ≤ i ≤ n − 1.Using (15) and ( 16) we get The fulfillment of the relations ( 2) is checked directly.
(2) Let us now show that the forbidden relations do not hold under the map Θ n .Indeed, we have: Thus the representation Θ n given by the formula (13) depends on the word a 1 (A, B) = w(A, B) ∈ F 2 = A, B , into which we substitute y i and y i+1 instead of A and B respectively, and a set of integers m = (m 1 , . . ., m n−1 ).To emphasize this dependence of the representation on w and m, we denote it Θ w,m n : (17) Θ w,m n (σ i ) : Θ w,m n (ρ i ) : where in the products The word w is called the defining word for the homomorphism Θ w,m n .In the particular case when m i = 0 for all i = 1, . . ., n − 1, we will write Θ w n : F V B n → Aut(F 2n ) assuming that (19) Θ w n (σ i ) : Note that the homomorphism (11) constructed in [20] can be represented as η n = Θ w n for w(A, B) = B.
Let β ∈ F V B n and x ∈ F 2n .Further, to simplify the notation, by β(x) we mean Θ w,m n (β)(x), where the word w and the set m are assumed to be clear from the context.
3. The kernel of homomorphism Θ w,m n and F V K n group In this section we show that the kernel of the homomorphism Θ w,m n lies in the intersection of the group of flat virtual pure braids and the group of flat virtual kure braids group F V K n defined below.
Consider the subgroup S n = σ 1 . . .σ n−1 of F V B n , which is isomorphic to the permutation group of an n-element set.The map π n : F V B n → S n defined on the generators σ i , ρ i according to the rule: is obviously a homomorphism.
Definition 2. Denote F V P n = Ker(π n ) and call it flat virtual pure braid group on n strands.
Similarly, the subgroup S ′ n = ρ 1 . . .ρ n−1 of F V B n is isomorphic to the permutation group of an n-element set, and the map ν n : F V B n → S ′ n defined on generators σ i , ρ i as follows: is also a homomorphism.
Definition 3. Denote F V K n = Ker(ν n ) and call it flat virtual kure braid group on n strands.
Here we use the term flat virtual kure braid since the term kure virtual braid group was used in [21] for kernel of the map π K : V B n → S n which is defined analogously to ν n : F V B n → S ′ .The group F V K n = Ker(ν n ) also was denoted by F H n in [20] since is the flat analog of the Rabenda's group H n from [22,Prop. 17].
Lemma 1. [20,Prop. 4] The group F V K n admits a presentation with generators x i,j , 1 ≤ i = j ≤ n and defining relations (20) x 2 i,j = 1, where different letters stand for different indices.
Corollary 1.The group F V K n is a Coxeter group with defining relations The following property is a generalization of the property established in [20,Prop. 9] for the word w(A, B) = B. Lemma 2. Let n ≥ 2. For any word w ∈ F 2 and any set of integers m = (m 1 , . . ., m n−1 ) , since all σ i act identically on y i .But then ν n (g) is the identity permutation of the set {y 1 , . . ., y n }, which by definition means g ∈ F V K n .
Next, we show that g ∈ F V P n .Denote by G the normal closure of the subgroup From the formulas (13) we can write out the action of Ψ w,m n on the generators of the group F V B n : Ψ w,m n (σ i ) : Now it is easy to see that the image of F V B n under the map Ψ w,m n is a permutation of the set {x 1 , . . ., x n }.It remains to note that if g ∈ Ker(Θ w,m n ), then g ∈ Ker(Ψ w,m n ) = F V P n .Since S ′ n ≤ F V B n , then the decomposition of F V B n = F V K n ⋊ S ′ n follows directly from the definition of F V K n .Considering the restriction of the homomorphism π n to F V K n , we obtain the homomorphism ξ n : we obtain the decomposition F V K n = X n ⋊S n .Thus, we have the following decomposition of the group of flat virtual braids: As it invented in [22], we denote Element λ i,j is presented geometrically in Figure 7, The group F V P n is generated by the elements λ i,j , 1 ≤ i < j ≤ n and the defining relations are: where i, j, k, l correspond to different indices.Let us consider the case n = 3 in more details.As shown in [22], where a = λ 2,3 λ 1,3 , b = λ 2,3 and c = λ 2,3 λ −1 1,2 .These elements presented geometrically in Figure 8. Theorem 2. We have the following decomposition where F 3 is a free group of rank 3 and Γ = x, y, u, v, p, q | xy = uv, vu = pq, qp = yx .
Proof.Consider the restriction of the homomorphism ν 3 : F V B 3 → S ′ 3 to F V P 3 .Let us denote it by ϕ : F V P 3 → S ′ n .Then X 3 = Ker(ϕ).To find the generators and relations of the X 3 group, we use the Reidemeisetr-Schreier rewriting process, see for example [23].Let us write out the system of Schreier representatives for Ker(ϕ) using the generators indicated in the presentation (24): T = {1, a, ab, c, cb, b}.For an element g, we denote its representative in T by g.Then the kernel Ker(ϕ) is generated by the following elements: Further, the relations taca −1 c −1 t −1 for t ∈ T must be rewritten in new generators.For example, for t = ab we get: The rest of the relations are found similarly.As a result, we get: It is now clear that the elements g, t generate Z 2 , the elements w, h, f generate F 3 , and the group generated by the elements x y, u, v, p and q we denote by Γ. Lemma 4. [20] Let G n be a subgroup of F V P n generated by the elements: Then the normal closure of G n in F V P n coincides with X n .
Let us describe the action of Θ w n on the generators indicated in Lemma 4.
Proof.Denote a free group of rank 2 by F 2 .By Lemma 6, it suffices to show that We write them down in terms of the generators of F V P 3 group, see (24): Let us prove that h 0 and h 1 generate F 2 .To do this, it suffices to show that there are no relations between them.Let ψ : F V P 3 → a, b be the homomorphism given by the mapping ψ(a) = a, ψ(b) = b and ψ(c) = 1.Denote h0 = ψ(h 0 ) and h1 = ψ(h 1 ).Then The elements h0 and h1 lie in the free group a, b .Hence the group h0 , h1 is either isomorphic to Z or isomorphic to F 2 .The first case means that h0 and h1 must be powers of the same element, i.But then g k = f w k f −1 = w k = h0 must be cyclically reduced, which is not the case.Thus h0 , h1 ∼ = F 2 and hence h 0 , h 1 ∼ = F 2 .
Corollary 2. Let n ≥ 3, then Ker(Θ w,m n ) contains a subgroup isomorphic to a free group of rank 2 for any integer tuple m = (m 1 , . . ., m n−1 ) and arbitrary defining word w(A, B) ∈ F 2 (A, B).
e. h0 = g(a, b) k and h1 = g(a, b) s for some word g(a, b) ∈ a, b and nonzero k, s ∈ Z.Let g(a, b) = f • w • f −1 , where w(a, b) is the cyclic reduced word in a, b .Then g s = f w s f −1 = h1 and since h1 is itself cyclically reduced, we get f = 1.