Multi-welded twin groups

Abstract

For $k\ge 1$ and $n\ge 2$, we introduce the multi-welded twin group $M_{k}W{T}_n$, a natural welded analogue of the multi-virtual twin group. We show that $M_{k}W{T}_n$ arises naturally as a quotient of the universal welded braid group $U{W}_n(k)$, placing it within the unified framework of universal virtual and welded braid-type groups. We establish natural quotient maps relating $M_{k}W{T}_n$ to the multi-virtual twin group $M_{k}V{T}_n$, the welded twin group $W{T}_n$, and the corresponding virtual and welded braid-type groups. Several structural properties of $M_{k}W{T}_n$ are obtained. In particular, we compute its abelianization, prove that its commutator subgroup is perfect for $n\ge 5$, and show that the symmetric group $S_n$ is its smallest non-abelian finite quotient. We also investigate the representation theory of $M_{k}W{T}_n$. In fact, we classify all non-trivial complex homogeneous $2$-local representations of $M_{k}W{T}_n$, showing that only one family survives under the additional twin and welded relations. Furthermore, we classify all non-trivial complex homogeneous $3$-local representations of $M_{2}W{T}_n$. We further investigate the reducibility and faithfulness properties of both the $2$-local and $3$-local representations.