The unrestricted virtual braid groups $UVB_n$

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