A group is called residually nilpotent if given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent. It is known that pure braid groups are residually nilpotent.


Consider the pure homotopy braid group which is a factor group of the pure braid group; namely, in the deformation of braid strands, each strand is allowed to self-intersect. Do we know if pure homotopy braid groups are residually nilpotent?



Are homotopy braid groups residually nilpotent?
The pure homotopy braid group is torsion-free nilpotent. This is Proposition 1.12 in N. Habegger and X.-S. Lin, The classification of links up to link homotopy, Journal of the American Mathematical Society 3 (1990), 389-419, where $\mathcal{A}(K)$ is the pure homotopy braid group. However, the authors do not use the terminology "pure homotopy braid group".


