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
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/

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