Virtual and universal braid groups, their quotients and representations (2107.03875v1)

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 $UB_n$.