On virtual singular braid groups

2022/07/28 by Ocampo, Oscar
#20F05 #20F36 #57K12 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2207.13885

Abstract

Let n≥ 2. In this paper we study some properties of the virtual singular braid group VSG_n. We define some numerical invariants for virtual singular braids arising from exponents of words in VSG_n and describe the kernel of these homomorphisms. We determine all possible homomorphisms, up to conjugation, from the virtual singular braid group VSG_n to the symmetric group S_n, and for the particular case n=2 we give a presentation (and a description) for the kernel in each case. For all possible homomorphisms we obtain a decomposition of VSG_n as a semi-direct product of the kernel of the homomorphism and the symmetric group. Finally, we show that there are some forbidden relations in VSG_n and then we introduce some quotients of it (e.g. the welded singular braid group, the unrestricted virtual singular braid group, among other related groups) and then we state for them similar results obtained for VSG_n.

Related

• Algebraic structures among virtual singular braids
• The virtual and universal braids
• The monoid structure of singular twisted virtual braids
• Virtual braids and permutations
• The unrestricted virtual braid groups UVB_n