Combinatorial properties of virtual braids

We determine the lower central series of the virtual braid group VBn and of the kernels of two different projections of VBn in Sn: the normal closure of the Artin braid group Bn, that we will denote by Hn, and the so-called virtual pure braid group VPn, which is related to Yang Baxter equation. We describe relations between Hn and VPn and we provide a connection between virtual pure braids and the finite type invariant theory for virtual knots defined by Goussarov, Polyak and Viro.