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 such that n5, m2 and nm. 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 /2×/2 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 tels que n5, m2 et nm. 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 /2×/2 et que VB n est à la fois hopfien et co-hofpien.

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Accepted:
Published online:
DOI: 10.5802/aif.3336
Classification: 20F36, 20E36
Keywords: virtual braid group, Bass–Serre theory, Artin group, symmetric group, amalgamated product
Mot clés : groupe de tresses virtuelles, théorie de Bass–Serre, groupe d’Artin, groupe symétrique, produit amalgamé

Bellingeri, Paolo 1; Paris, Luis 2

1 LMNO, UMR 6139, CNRS Université de Caen-Normandie 14000 Caen (France)
2 IMB, UMR 5584, CNRS Univ. Bourgogne Franche-Comté 21000 Dijon (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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/articles/10.5802/aif.3336/

[1] Artin, Emil Braids and permutations, Ann. Math., Volume 48 (1947), pp. 643-649

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

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

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

[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

[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 | Zbl

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

[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 | Zbl

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

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

[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

[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 | Zbl

[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

[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

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

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

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

[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, 46, Société Mathématique de France, 1977 | Zbl

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

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