The unrestricted virtual braid groups $UV{B}_n$

Abstract

Let $UV{B}_n$ and $UV{P}_n$ be the unrestricted virtual braid group and the unrestricted virtual pure braid group on n strands respectively. We study the groups $UV{B}_n$ and $UV{P}_n$, and our main results are as follows: for $n\ge 5$, we give a complete description, up to conjugation, to all possible homomorphisms from $UV{B}_n$ to the symmetric group $S_n$. For $n\ge 3$, we characterise all possible images of $UV{B}_n$, under a group homomorphism, to any finite group $G$. For $n\ge 5$, we prove that $UV{P}_n$ is a characteristic subgroup of $UV{B}_n$. In addition, we determine the automorphism group of $UV{P}_n$ and we prove that ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ is a subgroup of the outer automorphism group of $UV{B}_n$. Lastly, we show that $UV{B}_n$ and $UV{P}_n$ are residually finite and Hopfian but not co-Hopfian. We also remark that some of these results hold accordingly for the welded braid group $W{B}_n$ and we discuss about its automorphism group.

arXiv comment14 pages, to appear in Journal of Knot Theory and Its Ramifications