Commutator subgroups of welded braid groups

Abstract

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