Representations of virtual braids by automorphisms and virtual knot groups

In the present paper, a new representation of the virtual braid group VBn into the automorphism group of free product of the free group and free abelian group is constructed. This representation generalizes the previously constructed ones. The fact that the previously known representations are not faithful for n ≥ 4 is verified. Using representations of VBn, a virtual link group is defined. Also representations of the welded braid group WBn are constructed and the welded link group is defined.