Virtual braids and permutations  [ Tresses virtuelles et permutations ]
Annales de l'Institut Fourier, à paraître, 22 p.

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.

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.

Reçu le : 2018-10-22
Accepté le : 2019-05-21
Classification : 20F36,  20E36
Mots clés: groupe de tresses virtuelles, théorie de Bass–Serre, groupe d’Artin, groupe symétrique, produit amalgamé
@unpublished{AIF_0__0_0_A20_0,
     author = {Bellingeri, Paolo and Paris, Luis},
     title = {Virtual braids and permutations},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Bellingeri, Paolo; Paris, Luis. Virtual braids and permutations. Annales de l'Institut Fourier, à paraître, 22 p.

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