Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Commutator Subgroups of Virtual and Welded Braid Groups   Let $VB_n$, resp. $WB_n$ denote the virtual, resp. welded, braid group on $n$
strands. We study their commutator subgroups $VB_n' = [VB_n, VB_n]$ and, $WB_n'
= [WB_n, WB_n]$ respectively. We obtain a set of generators and defining
relations for these commutator subgroups. In particular, we prove that $VB_n'$
is finitely generated if and only if $n \geq 4$, and $WB_n'$ is finitely
generated for $n \geq 3$. Also we prove that $VB_3'/VB_3'' =\mathbb{Z}_3 \oplus
\mathbb{Z}_3 \oplus\mathbb{Z}_3 \oplus \mathbb{Z}^{\infty}$, $VB_4' / VB_4'' =
\mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_3$, $WB_3'/WB_3'' =
\mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus\mathbb{Z}_3 \oplus \mathbb{Z},$
$WB_4'/WB_4'' = \mathbb{Z}_3,$ and for $n \geq 5$ the commutator subgroups
$VB_n'$ and $WB_n'$ are perfect, i.e. the commutator subgroup is equal to the
second commutator subgroup.

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The categories of this paper are: nucl-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: q-fin
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: chem-ph
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The categories of this paper are: cs