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Commutator subgroups of singular braid groups

Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Kelambakkam 603103, Tamil Nadu, India

and

Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, SAS Nagar, P. O. Manauli, Punjab 140306, India

E-mail Address: krishnendu@iisermohali.ac.in

E-mail Address: krishnendug@gmail.com

Corresponding author.

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https://doi.org/10.1142/S021821652250033XCited by:0 (Source: Crossref)

Abstract

The singular braids with $n$ strands, $n \geq 3$ , were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by $S G_{n}$ . There has been another generalization of braid groups, denoted by $G V B_{n}$ , $n \geq 3$ , which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group $G V B_{n}$ simultaneously generalizes the classical braid group, as well as the virtual braid group on $n$ strands.

We investigate the commutator subgroups $S G_{n}^{'}$ and $G V B_{n}^{'}$ of these generalized braid groups. We prove that $S G_{n}^{'}$ is finitely generated if and only if $n \geq 5$ , and $G V B_{n}^{'}$ is finitely generated if and only if $n \geq 4$ . Further, we show that both $S G_{n}^{'}$ and $G V B_{n}^{'}$ are perfect if and only if $n \geq 5$ .

Keywords:

AMSC: 20F36, 20F12, 20F05

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