The unrestricted virtual braid groups $UVB_n$

TL;DR

This paper investigates the algebraic structure and automorphisms of the unrestricted virtual braid groups and pure braid groups, providing classifications of homomorphisms, automorphisms, and residual properties, with implications for related welded braid groups.

Contribution

It offers a comprehensive description of homomorphisms from $UVB_n$ to symmetric groups, characterizes images in finite groups, and determines automorphism groups, advancing understanding of these complex algebraic structures.

Findings

Complete classification of homomorphisms to $S_n$ for $n extgreater=5$

Characterization of images in arbitrary finite groups for $n extgreater=3$

Determination of automorphism groups and residual properties

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.