Virtual and universal braid groups, their quotients and representations

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 $B_n\to {\mathrm{GL}}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm 1}]\right)$, $V{B}_n\to {\mathrm{GL}}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm 1},t_1^{\pm 1},t_2^{\pm 1},\dots ,t_{n-1}^{\pm 1}]\right)$ which are connected with the famous Lawrence-Bigelow-Krammer representation. It turns out that these representations are faithful representations of crystallographic groups $B_n/P_n^{\prime }$, $V{B}_n/V{P}_n^{\prime }$, respectively. Using these representations we study certain properties of the groups $B_n/P_n^{\prime }$, $V{B}_n/V{P}_n^{\prime }$. Moreover, we construct new representations and decompositions of universal braid groups $U{B}_n$.

29 pages