Corpus ID: 235765706

Virtual and universal braid groups, their quotients and representations

  title={Virtual and universal braid groups, their quotients and representations},
  author={Valerij G. Bardakov and I. Emel'yanenkov and Maxim Ivanov and T. A. Kozlovskaya and Timur Nasybullov and A. Vesnin},
In the present paper we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representations Bn → GLn(n−1)/2 ( Z[t] ) , V Bn → GLn(n−1)/2 ( Z[t, t 1 , t ±1 2 , . . . , t ±1 n−1] ) which are connected with the famous Lawrence-Bigelow-Krammer representation. It turns out that these representations are faithful representations of crystallographic groups Bn/P ′ n, V Bn/V P ′ n, respectively. Using these representations… Expand
1 Citations
Virtual braid groups, virtual twin groups and crystallographic groups
Let n ≥ 2. Let V Bn (resp. V Pn) be the virtual braid group (resp. the pure virtual braid group), and let V Tn (resp. PV Tn) be the virtual twin group (resp. the pure virtual twin group). Let Π beExpand


A quotient of the Artin braid groups related to crystallographic groups
Abstract Let n ≥ 3 . In this paper, we study the quotient group B n / [ P n , P n ] of the Artin braid group B n by the commutator subgroup of its pure Artin braid group P n . We show that B n / [ PExpand
Combinatorial properties of virtual braids
Abstract We determine the lower central series of the virtual braid group VB n and of the kernels of two different projections of VB n in S n : the normal closure of the Artin braid group B n , thatExpand
Representations of virtual braids by automorphisms and virtual knot groups
In the present paper the representation of the virtual braid group $VB_n$ into the automorphism group of free product of the free group and free abelian group is constructed. This representationExpand
Motivated by the recent concept of a pseudosymmetric braided monoidal category, we define the pseudosymmetric group PS n to be the quotient of the braid group B n by the relations σ i σ ―1 i+1 σ i =Expand
Braid groups are linear
The braid groups B_n can be defined as the mapping class group of the n-punctured disc. The Lawrence-Krammer representation of the braid group B_n is the induced action on a certain twisted secondExpand
On the Pure Virtual Braid Group PV3
We investigate various properties of the pure virtual braid group PV3. Out of its presentation, we get a free product decomposition of PV3. As a consequence, we show that PV3 is residually torsionExpand
A Representation of Virtual Braids by Automorphisms
We study a representation of the virtual braid group VBn into the automorphism group of a free product of a free group and a free Abelian group, proposed by S. Kamada. It is proved that the givenExpand
Braid groups are
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The virtual and universal braids
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 thatExpand
On the Burau representation for $n=4$
The problem of faithfulness of the (reduced) Burau representation for $n =4$ is known to be equivalent to the problem of whether certain two matrices A and B generate a free group of rank two. InExpand