Virtual and universal braid groups, their quotients and representations

TL;DR

This paper investigates quotients of braid and virtual braid groups, constructs linear representations related to the Lawrence-Bigelow-Krammer representation, and explores their properties and decompositions, revealing faithfulness and new structural insights.

Contribution

It introduces new linear representations of braid and virtual braid groups connected to known representations, demonstrating their faithfulness and analyzing group properties.

Findings

Representations are faithful for certain quotient groups.

Constructed new representations and decompositions of universal braid groups.

Analyzed properties of quotients of braid groups and virtual braid groups.

Abstract

In the present paper we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representations $B_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}]\right)$, $VB_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}, t_1^{\pm1},t_2^{\pm1},\ldots, t_{n-1}^{\pm1}]\right)$ which are connected with the famous Lawrence-Bigelow-Krammer representation. It turns out that these representations are faithful representations of crystallographic groups $B_n/P_n'$, $VB_n/VP_n'$, respectively. Using these representations we study certain properties of the groups $B_n/P_n'$, $VB_n/VP_n'$. Moreover, we construct new representations and decompositions of universal braid groups $UB_n$.