Characteristic subgroups and the R_-property for virtual braid groups

Let n 2. Let VBₙ (resp. VPₙ) denote the virtual braid group (resp. virtual pure braid group), let WBₙ (resp. WPₙ) denote the welded braid group (resp. welded pure braid group) and let UVBₙ (resp. UVPₙ) denote the unrestricted virtual braid group (resp. unrestricted virtual pure braid group). In the first part of this paper we prove that, for n 4, the group VPₙ and for n 3 the groups WPₙ and UVPₙ are characteristic subgroups of VBₙ, WBₙ and UVBₙ, respectively. In the second part of the paper we show that, for n 2, the virtual braid group VBₙ, the unrestricted virtual pure braid group UVPₙ, and the unrestricted virtual braid group UVBₙ have the R_-property. As a consequence of the technique used for few strings we also prove that, for n=2, 3, 4, the welded braid group WBₙ has the R_-property and that for n=2 the corresponding pure braid groups have the R_-property. On the other hand for n 3 it is unknown if the R_-property holds or not for the virtual pure braid group VPₙ and the welded pure braid group WPₙ.