Virtual and universal braid groups, their quotients and representations

Abstract

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} .