Characteristic subgroups and the R∞-property for virtual braid groups.

Let n ≥ 2. Let V B n (resp. V P n) denote the virtual braid group (resp. virtual pure braid group), let W B n (resp. W P n) denote the welded braid group (resp. welded pure braid group) and let U V B n (resp. U V P n) 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 V P n and for n ≥ 3 the groups W P n and U V P n are characteristic subgroups of V B n , W B n and U V B n , respectively. In the second part of the paper we show that, for n ≥ 2 , the virtual braid group V B n , the unrestricted virtual pure braid group U V P n , and the unrestricted virtual braid group U V B n 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 W B n 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 V P n and the welded pure braid group W P n. [ABSTRACT FROM AUTHOR]