A simple solution to the word problem for virtual braid groups

Bellingeri, Paolo; Cruz, Bruno A. Cisneros de la; Paris, Luis

doi:10.2140/pjm.2016.283.271

Cited by 7 publications

(10 citation statements)

References 13 publications

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“…It is denoted by KB n . It is an Artin group, hence we can use tools from the theory of Artin groups to study it and get results on VB n itself (see for example Godelle-Paris [14] or Bellingeri-Cisneros de la Cruz-Paris [8]). This group will play a prominent role in our study.…”

Section: τ N−1 and Relationsmentioning

confidence: 99%

“…Despite their interest in low-dimensional topology, the virtual braid groups are poorly understood from a combinatorial point of view. Among the known results for these groups there are solutions to the word problem in Bellingeri-Cisneros de la Cruz-Paris [8] and in Chterental [10] and the calculation of some terms of its lower central series in , and some other results but not many more. For example, it is not known whether these groups have a solution to the conjugacy problem or whether they are linear.…”

Section: Introductionmentioning

confidence: 99%

“…Our study is totally independent from the works of Dyer-Grossman [13], Lin [18,19] and Castel [9] cited above and all similar works on braid groups. Our viewpoint/strategy is also new for virtual braid groups, although some aspects are already present in Bellingeri-Cisneros de la Cruz-Paris [8]. Our idea/main-contribution consists on first observing that the virtual braid group VB n decomposes as a semi-direct product VB n = KB n S n of an Artin group KB n by the symmetric group S n , and secondly and mainly on making a deep study of the action of S n on KB n .…”

Section: Introductionmentioning

confidence: 99%

“…The group VB 2 is isomorphic to Z * Z/2Z. On the other hand, fairly precise studies of the combinatorial structure of VB 3 can be found in , Bardakov-Mikhailov-Vershinin-Wu [5] and Bellingeri-Cisneros de la Cruz-Paris [8]. From this one can probably determine (with some difficulties) all the homomorphisms from VB n to S m , from S n to VB m , and from VB n to VB m , for n m and n = 2, 3.…”

Section: Introductionmentioning

confidence: 99%

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Virtual braids and permutations

Bellingeri¹,

Paris²

2020

Annales De l'Institut Fourier

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Section: τ N−1 and Relationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Virtual braids and permutations

Bellingeri¹,

Paris²

2020

Annales De l'Institut Fourier

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“…In particular KVB n is also of type F, therefore it is torsion free and it has a solution to the word problem. Since KVB n is of finite index in VB n , we can therefore use KVB n to solve the word problem in VB n (see Bellingeri-Cisneros-de-la-Cruz-Paris [5]).…”

Section: Introductionmentioning

confidence: 99%

Virtual Artin groups

Bellingeri¹,

Paris²,

Thiel³

2021

Preprint

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Starting from the observation that the standard presentation of a virtual braid group mixes the standard presentation of the corresponding braid group with the standard presentation of the corresponding symmetric group and some mixed relations that mimic the action of the symmetric group on its root system, we define a virtual Artin group VA[Γ] of a Coxeter graph Γ mixing the standard presentation of the Artin group A[Γ] with the standard presentation of the Coxeter group W[Γ] and some mixed relations that mimic the action of W[Γ] on its root system. By definition we have two epimorphisms π K : VA[Γ] → W[Γ] and π P : VA[Γ] → W[Γ] whose kernels are denoted by KVA[Γ] and PVA[Γ] respectively. We calculate presentations for these two subgroups. In particular KVA[Γ] is an Artin group. We prove that the center of any virtual Artin group is trivial. In the case where Γ is of spherical type or of affine type, we show that each free of infinity parabolic subgroup of KVA[Γ] is also of spherical type or of affine type, and we show that VA[Γ] has a solution to the word problem. In the case where Γ is of spherical type we show that KVA[Γ] satisfies the K(π, 1) conjecture and we infer the cohomological dimension of KVA[Γ] and the virtual cohomological dimension of VA[Γ]. In the case where Γ is of affine type we determine upper bounds for the cohomological dimension of KVA[Γ] and for the virtual cohomological dimension of VA [Γ].

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Virtual Artin groups

Bellingeri

Paris

Thiel

2022

Proceedings of London Math Soc

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Starting from the observation that the standard presentation of a virtual braid group mixes the standard presentation of the corresponding braid group with the standard presentation of the corresponding symmetric group and some mixed relations that mimic the action of the symmetric group on its root system, we define a virtual Artin group VA[Γ] of a Coxeter graph Γ mixing the standard presentation of the Artin group 𝐴[Γ] with the standard presentation of the Coxeter group 𝑊[Γ] and some mixed relations that mimic the action of 𝑊[Γ] on its root system. By definition, we have two epimorphisms 𝜋 𝐾 ∶ VA[Γ] → 𝑊[Γ] and 𝜋 𝑃 ∶ VA[Γ] → 𝑊[Γ] whose kernels are denoted by KVA[Γ] and PVA[Γ],respectively. We calculate presentations for these two subgroups. In particular, KVA[Γ] is an Artin group.We prove that the center of any virtual Artin group is trivial. In the case where Γ is of spherical type or of affine type, we show that each free of infinity parabolic subgroup of KVA[Γ] is also of spherical type or of affine type, and we show that VA[Γ] has a solution to the word problem. In the case where Γ is of spherical type we show that KVA[Γ] satisfies the 𝐾(𝜋, 1) conjecture and we infer the cohomological dimension of KVA[Γ] and the virtual cohomological dimension of VA[Γ]. In the case where Γ is of affine type we determine upper bounds for the cohomological dimension of

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A simple solution to the word problem for virtual braid groups

Abstract: International audienceWe show a simple and easily implementable solution to the word problem for virtual braid groups

Cited by 7 publications

References 13 publications

Virtual braids and permutations

Virtual braids and permutations

Virtual Artin groups

Virtual Artin groups

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