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

Abstract

Let $n\geq 2$. Let $VB_n$ (resp. $VP_n$) denote the virtual braid group (resp. virtual pure braid group), let $WB_n$ (resp. $WP_n$) denote the welded braid group (resp. welded pure braid group) and let $UVB_n$ (resp. $UVP_n$) denote the unrestricted virtual braid group (resp. unrestricted virtual pure braid group). In the first part of this paper we prove that, for $n\geq 4$, the group $VP_n$ and for $n\geq 3$ the groups $WP_n$ and $UVP_n$ are characteristic subgroups of $VB_n$, $WB_n$ and $UVB_n$, respectively. In the second part of the paper we show that, for $n\geq 2$, the virtual braid group $VB_n$, the unrestricted virtual pure braid group $UVP_n$, and the unrestricted virtual braid group $UVB_n$ 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 $WB_n$ 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 $n\geq 3$ it is unknown if the R$_\infty$-property holds or not for the virtual pure braid group $VP_n$ and the welded pure braid group $WP_n$.