Virtual and universal braid groups, their quotients and representations

2021/07/08 by Bardakov, V., Emel'yanenkov, I., Ivanov, M. +3 · 1 citation
#20F29 #20F36 #57M25 #57M27 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2107.03875

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→ \rm GL_n(n-1)/2(ℤ[t^±1]), VB_n→ \rm GL_n(n-1)/2(ℤ[t^±1, t₁^±1,t₂^±1,…, t_n-1^±1]) 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.

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• Multi-virtual braid groups

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