The \(D_ n\) generalized pure braid group (Q1176345)

The author carries out a detailed analysis of a generalized pure braid group \(ID_ k\) associated with the Weyl group of type \(D_ k\). A natural system of generators is constructed and the corresponding presentation is derived. This presentation exhibits \(ID_ k\) as a semidirect product of \(ID_{k-1}\) by a group \(H_ k\) isomorphic to the free product of \(k-1\) copies of the free abelian group of rank two. It is then shown that this product is ``almost direct'', in the sense that \(ID_{k-1}\) acts trivially on the abelianization of \(H_ k\). This result is then used to show that \(ID_ k\) is residually torsion-free nilpotent.