LAUSR

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 → GL n ⁢ ( n - 1 ) / 2 ⁢ ( Z ⁢ [ t ± 1 ] ) B_{n}\to\mathrm{GL}_{n(n-1)/2}(\mathbb{Z}[t^{\pm 1}]) , VB n → GL n ⁢ ( n - 1 ) / 2 ⁢ ( Z ⁢ [ t ± 1 , t 1 ± 1 , t 2 ± 1 , … , t n - 1 ± 1 ] ) \mathrm{VB}_{n}\to\mathrm{GL}_{n(n-1)/2}(\mathbb{Z}[t^{\pm 1},t_{1}^{\pm 1},t_{2}^{\pm 1},\ldots,t_{n-1}^{\pm 1}]) which are connected with the famous Lawrence–Bigelow–Krammer representation. It turns out that these representations induce faithful representations of the crystallographic groups B n / P n ′ B_{n}/P_{n}^{\prime} , VB n / VP n ′ \mathrm{VB}_{n}/\mathrm{VP}_{n}^{\prime} , respectively. Using these representations we study certain properties of the groups B n / P n ′ B_{n}/P_{n}^{\prime} , VB n / VP n ′ \mathrm{VB}_{n}/\mathrm{VP}_{n}^{\prime} . Moreover, we construct new representations and decompositions of the universal braid groups UB n \mathrm{UB}_{n} .