Characteristic subgroups and the R${}_{\infty }$-property for virtual braid groups

Abstract

Let $n\ge 2$. 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 $UV{B}_n$ (resp. $UV{P}_n$) denote the unrestricted virtual braid group (resp. unrestricted virtual pure braid group). In the first part of this paper we prove that, for $n\ge 4$, the group $V{P}_n$ and for $n\ge 3$ the groups $W{P}_n$ and $UV{P}_n$ are characteristic subgroups of $V{B}_n$, $W{B}_n$ and $UV{B}_n$, respectively. In the second part of the paper we show that, for $n\ge 2$, the virtual braid group $V{B}_n$, the unrestricted virtual pure braid group $UV{P}_n$, and the unrestricted virtual braid group $UV{B}_n$ have the R${}_{\infty }$-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${}_{\infty }$-property and that for $n=2$ the corresponding pure braid groups have the R${}_{\infty }$-property. On the other hand for $n\ge 3$ it is unknown if the R${}_{\infty }$-property holds or not for the virtual pure braid group $V{P}_n$ and the welded pure braid group $W{P}_n$.

26 pages