On representations of virtual braid group and groups of virtual links

We introduce some representation $\psi$ of the virtual braid group $VB_n$ into the automorphism group $Aut(F_{n,2n+1})$ of a free product $F_{n,2n+1} = F_n * \mathbb{Z}^{2n+1}$, where $F_n$ is a free group and $\mathbb{Z}^{2n+1}$ is a free abelian group. This representation generalizes some other representations. In particular, the representation $\varphi_0 : VB_n \longrightarrow Aut(F_{n})$ defined in [1]; the representation $\varphi_1 : VB_n \longrightarrow Aut(F_{n+1})$ defined in [2], [3] (see also, [4]); the representation $\varphi_2 : VB_n \longrightarrow Aut(F_{n,n+1})$ defined in [5]; the representation $\varphi_3 : VB_n \longrightarrow Aut(F_{n,2})$ defined in [6]. On the other hand the Artin representation is faithful. It is interesting to construct a representation which is an extension of it.
Theorem 1. {\sl There is a representation $VB_n \longrightarrow Aut(F_{n,n})$ which is an extension of Artin representation and in some sense is equivalent to the representation $\psi$.}
From the result of O. Chterental [7] follows that for $n > 3$ the representations $\varphi_1$, $\varphi_2$ and $\varphi_3$ have non-trivial kernels. Analogous question for $\psi$ is opened.
Using any of the representation $\psi, \varphi_0, \varphi_1, \varphi_2, \varphi_3$ one can defines a group $G_{\psi}(L)$, $G_{\varphi_0}(L)$, $G_{\varphi_1}(L)$, $G_{\varphi_2}(L)$, $G_{\varphi_3}(L)$ of a virtual link $L$. A connection between these groups gives
Theorem 2. {\sl The groups $G_{\varphi_0}(L)$, $G_{\varphi_1}(L)$, $G_{\varphi_2}(L)$, $G_{\varphi_3}(L)$ are homomorphic images of the group $G_{\psi}(L)$. If $L$ is a virtual knot, then we have isomorphisms $G_{\psi}(L) \cong G_{\varphi_1}(L) \cong G_{\varphi_2}(L) \cong G_{\varphi_3}(L)$.}
The talk is based on the joint work with M. V. Meshchadim and Yu. A. Mikhalchishina [8]. The author is partially supported by the Laboratory of Quantum Topology of Chelyabinsk State University (Russian Federation government grant 14.Z50.31.0020) and RFBR grant 16-01-00414 and RNF grant 16-41-02006
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