Let be the virtual braid group on strands and let be the symmetric group on letters. Let such that , and . We determine all possible homomorphisms from to , from to and from to . As corollaries we get that is isomorphic to and that is both, Hopfian and co-Hofpian.
Soient le groupe de tresses virtuelles à brins et le groupe symétrique de l’ensemble à éléments. Soient tels que , et . Nous déterminons tous les homomorphismes de dans , de dans et de dans . Comme corollaires nous obtenons que est isomorphe à et que est à la fois hopfien et co-hofpien.
Accepted:
Published online:
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
@article{AIF_2020__70_3_1341_0, author = {Bellingeri, Paolo and Paris, Luis}, title = {Virtual braids and permutations}, journal = {Annales de l'Institut Fourier}, pages = {1341--1362}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {70}, number = {3}, year = {2020}, doi = {10.5802/aif.3336}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3336/} }
TY - JOUR AU - Bellingeri, Paolo AU - Paris, Luis TI - Virtual braids and permutations JO - Annales de l'Institut Fourier PY - 2020 SP - 1341 EP - 1362 VL - 70 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3336/ DO - 10.5802/aif.3336 LA - en ID - AIF_2020__70_3_1341_0 ER -
%0 Journal Article %A Bellingeri, Paolo %A Paris, Luis %T Virtual braids and permutations %J Annales de l'Institut Fourier %D 2020 %P 1341-1362 %V 70 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3336/ %R 10.5802/aif.3336 %G en %F 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/articles/10.5802/aif.3336/
[1] Braids and permutations, Ann. Math., Volume 48 (1947), pp. 643-649
[2] Theory of braids, Ann. Math., Volume 48 (1947), pp. 101-126
[3] The virtual and universal braids, Fundam. Math., Volume 184 (2004), pp. 1-18
[4] Combinatorial properties of virtual braids, Topology Appl., Volume 156 (2009) no. 6, pp. 1071-1082
[5] On the pure virtual braid group , Commun. Algebra, Volume 44 (2016) no. 3, pp. 1350-1378
[6] Groups and Lie algebras corresponding to the Yang–Baxter equations, J. Algebra, Volume 305 (2006) no. 2, pp. 742-764 | Zbl
[7] Braid groups and the co-Hopfian property, J. Algebra, Volume 303 (2006) no. 1, pp. 275-294 | Zbl
[8] A simple solution to the word problem for virtual braid groups, Pacific J. Math., Volume 283 (2016) no. 2, pp. 271-287 | Zbl
[9] Geometric representations of the braid groups, Astérisque, 378, Société Mathématique de France, 2016 | Zbl
[10] Virtual braids and virtual curve diagrams, J. Knot Theory Ramifications, Volume 24 (2015) no. 13, 1541001, 24 pages | Zbl
[11] 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] Virtual braids from a topological viewpoint, J. Knot Theory Ramifications, Volume 24 (2015) no. 6, 1550033, 36 pages | Zbl
[13] The automorphism groups of the braid groups, Am. J. Math., Volume 103 (1981) no. 6, pp. 1151-1169
[14] 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] Braid presentation of virtual knots and welded knots, Osaka J. Math., Volume 44 (2007) no. 2, pp. 441-458
[16] Virtual knot theory, Eur. J. Comb., Volume 20 (1999) no. 7, pp. 663-690 | Zbl
[17] The homotopy type of complex hyperplane complements, Ph. D. Thesis, University of Nijmegen (The Netherlands) (1983)
[18] Artin braids and related groups and spaces, Itogi Nauki Tekh., Ser. Algebra Topologiya Geom., Volume 17 (1979), pp. 159-227
[19] Braids and permutations (2004) (https://arxiv.org/abs/math/0404528)
[20] Arbres, amalgames, , Astérisque, 46, Société Mathématique de France, 1977 | Zbl
[21] On homology of virtual braids and Burau representation, J. Knot Theory Ramifications, Volume 10 (2001) no. 5, pp. 795-812
Cited by Sources: