no. 3
Virtual braids and permutations
Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 1341-1362.

Let VB n be the virtual braid group on n strands and let 𝔖 n be the symmetric group on n letters. Let n,m∈ℕ such that n≥5, m≥2 and n≥m. We determine all possible homomorphisms from VB n to 𝔖 m , from 𝔖 n to VB m and from VB n to VB m . As corollaries we get that Out(VB n ) is isomorphic to ℤ/2ℤ×ℤ/2ℤ and that VB n is both, Hopfian and co-Hofpian.

Soient VB n le groupe de tresses virtuelles à n brins et 𝔖 n le groupe symétrique de l’ensemble à n éléments. Soient n,m∈ℕ tels que n≥5, m≥2 et n≥m. Nous déterminons tous les homomorphismes de VB n dans 𝔖 m , de 𝔖 n dans VB m et de VB n dans VB m . Comme corollaires nous obtenons que Out(VB n ) est isomorphe à ℤ/2ℤ×ℤ/2ℤ et que VB n est à la fois hopfien et co-hofpien.

Received: 2018-10-22
Accepted: 2019-05-21
Published online: 2020-06-26
DOI: https://doi.org/10.5802/aif.3336
Classification: 20F36,  20E36
Keywords: virtual braid group, Bass–Serre theory, Artin group, symmetric group, amalgamated product 
@article{AIF_2020__70_3_1341_0,
     author = {Bellingeri, Paolo and Paris, Luis},
     title = {Virtual braids and permutations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {3},
     year = {2020},
     pages = {1341-1362},
     doi = {10.5802/aif.3336},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2020__70_3_1341_0/}
}
Bellingeri, Paolo; Paris, Luis. Virtual braids and permutations. Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 1341-1362. doi : 10.5802/aif.3336. https://aif.centre-mersenne.org/item/AIF_2020__70_3_1341_0/

[1] Artin, Emil Braids and permutations, Ann. Math., Volume 48 (1947), pp. 643-649 | Article | MR 20989 | Zbl 0030.17802

[2] Artin, Emil Theory of braids, Ann. Math., Volume 48 (1947), pp. 101-126 | Article | MR 19087

[3] Bardakov, Valerij G. The virtual and universal braids, Fundam. Math., Volume 184 (2004), pp. 1-18 | Article | MR 2128039 | Zbl 1078.20036

[4] Bardakov, Valerij G.; Bellingeri, Paolo Combinatorial properties of virtual braids, Topology Appl., Volume 156 (2009) no. 6, pp. 1071-1082 | Article | MR 2493369 | Zbl 1196.20044

[5] Bardakov, Valerij G.; Mikhailov, Roman; Vershinin, Vladimir V.; Wu, Jie On the pure virtual braid group PV 3 , Commun. Algebra, Volume 44 (2016) no. 3, pp. 1350-1378 | Article

[6] Bartholdi, Laurent; Enriquez, Benjamin; Etingof, Pavel; Rains, Eric Groups and Lie algebras corresponding to the Yang–Baxter equations, J. Algebra, Volume 305 (2006) no. 2, pp. 742-764 | Article | MR 2266850 | Zbl 1169.17004

[7] Bell, Robert W.; Margalit, Dan Braid groups and the co-Hopfian property, J. Algebra, Volume 303 (2006) no. 1, pp. 275-294 | Article | MR 2253663 | Zbl 1110.20028

[8] Bellingeri, Paolo; Cisneros de la Cruz, Bruno A.; Paris, Luis A simple solution to the word problem for virtual braid groups, Pacific J. Math., Volume 283 (2016) no. 2, pp. 271-287 | Article | MR 3519103 | Zbl 1375.20042

[9] Castel, Fabrice Geometric representations of the braid groups, Astérisque, Volume 378, Société Mathématique de France, 2016 | MR 3499033 | Zbl 06590103

[10] Chterental, Oleg Virtual braids and virtual curve diagrams, J. Knot Theory Ramifications, Volume 24 (2015) no. 13, 1541001, 24 pages | MR 3434540 | Zbl 1337.57013

[11] Crisp, John; Paris, Luis The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group, Invent. Math., Volume 145 (2001) no. 1, pp. 19-36 | Article | MR 1839284 | Zbl 1002.20021

[12] Cisneros de la Cruz, Bruno A. Virtual braids from a topological viewpoint, J. Knot Theory Ramifications, Volume 24 (2015) no. 6, 1550033, 36 pages | MR 3358444 | Zbl 1327.57004

[13] Dyer, Joan L.; Grossman, Edna K. The automorphism groups of the braid groups, Am. J. Math., Volume 103 (1981) no. 6, pp. 1151-1169 | Article | MR 636956 | Zbl 0476.20026

[14] Godelle, Eddy; Paris, Luis K(π,1) and word problems for infinite type Artin–Tits groups, and applications to virtual braid groups, Math. Z., Volume 272 (2012) no. 3-4, pp. 1339-1364 | Article | MR 2995171 | Zbl 1300.20045

[15] Kamada, Seiichi Braid presentation of virtual knots and welded knots, Osaka J. Math., Volume 44 (2007) no. 2, pp. 441-458 | MR 2351010 | Zbl 1147.57008

[16] Kauffman, Louis H. Virtual knot theory, Eur. J. Comb., Volume 20 (1999) no. 7, pp. 663-690 | Article | MR 1721925 | Zbl 0938.57006

[17] van der Lek, H. The homotopy type of complex hyperplane complements (1983) (Ph. D. Thesis)

[18] Lin, Vladimir J. Artin braids and related groups and spaces, Itogi Nauki Tekh., Ser. Algebra Topologiya Geom., Volume 17 (1979), pp. 159-227

[19] Lin, Vladimir J. Braids and permutations (2004) (https://arxiv.org/abs/math/0404528)

[20] Serre, Jean-Pierre Arbres, amalgames, SL 2 , Astérisque, Volume 46, Société Mathématique de France, 1977 | Numdam | MR 476875 | Zbl 0369.20013

[21] Vershinin, Vladimir V. On homology of virtual braids and Burau representation, J. Knot Theory Ramifications, Volume 10 (2001) no. 5, pp. 795-812 | Article | MR 1839703 | Zbl 0997.57020