Combinatorial properties of virtual braids

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