# Commutator Subgroups of Welded Braid Groups

## TL;DR

This paper studies the algebraic properties of commutator subgroups of welded braid groups, proving finite generation, Hopfian property, and conditions for perfectness, along with presentations for related groups.

## Contribution

It provides new results on the structure and properties of commutator subgroups of welded and flat welded braid groups, including finite presentations and criteria for perfectness.

## Key findings

- $WB_n'$ is finitely generated and Hopfian.
- $WB_n'$ is perfect if and only if $n \\geq 5$.
- Finite presentations for $FWB_n'$ are computed.

## Abstract

Let $WB_n$ be the welded (or loop) braid group on n strands, $n \geq 3$. We investigate commutator subgroup of $WB_n$. We prove that the commutator subgroup $WB_n'$ is finitely generated and Hopfian. We show that $WB_n'$ is perfect if and only if $n \geq 5$. We also compute finite presentation for $FWB_n'$, the commutator subgroup of the flat welded braid group $FWB_n$. Along the way, we investigate adorability of these groups.