Virtual braids and permutations

Abstract

Let VB${}_n$ be the virtual braid group on $n$ strands and let ${\mathfrak{S}}_n$ be the symmetric group on $n$ letters. Let $n,m\in \mathbb{N}$ such that $n\ge 5$, $m\ge 2$ and $n\ge m$. We determine all possible homomorphisms from VB${}_n$ to ${\mathfrak{S}}_m$, from ${\mathfrak{S}}_n$ to VB${}_m$ and from VB${}_n$ to VB${}_m$. As corollaries we get that Out(VB${}_n$) is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ and that VB${}_n$ is both Hopfian and co-Hofpian.