Forbidden relations in universal virtual braid groups

Abstract

We study natural automorphisms of the universal virtual braid group $U{V}_n(k)$. These automorphisms induce commuting involutions in the outer automorphism group and generate a subgroup isomorphic to ${\mathbb{Z}}_2^k\times {\mathbb{Z}}_2$. We then show that the two one-forbidden quotients of $U{V}_n(k)$ are isomorphic. Furthermore, we introduce the universal unrestricted virtual braid group $UU{V}_n(k)$ obtained by imposing simultaneously the two forbidden relations, and derive several structural properties inherited from the universal setting. Since the multi-virtual braid group $M_{k}V{B}_n$ is a quotient of $U{V}_n(k)$, the corresponding results for $M_{k}V{B}_n$ follow as consequences. In particular, for $k=1$ we prove that the quotients of $V{B}_n$ by the two forbidden relations are isomorphic and obtain structural properties for the unrestricted virtual braid group.

13 pages. Comments are welcome