# Are homotopy braid groups residually nilpotent?

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?

• Braid groups are not residually nilpotent (for $n\ge 5$, $[B_5,B_5]$ is perfect as far as I remember); you probably mean pure braid groups (which are indeed residually [torsion-free nilpotent]). – YCor Nov 14 '14 at 16:36
• Yes, @YCor, you are right!! I need to edit my question. Then what can we say about pure homotopy braid groups? – Zuriel Nov 14 '14 at 16:50

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".

• I guess you mean "residually torsion-free nilpotent" – YCor Apr 16 '16 at 17:22
• @YCor. I did mean "torsion-free nilpotent". – Ed Formanek Nov 29 '17 at 17:59

It was shown by M. Falk and R. Randell that the pure braid groups $P_n$ (and, more generally, fundamental groups of complements of fiber-type arrangements) are residually nilpotent (see Theorem 2.6 from Pure braid groups and products of free groups, in: Braids (Santa Cruz, CA, 1986), 217–228, Contemp. Math., 78, Amer. Math. Soc., Providence, RI, 1988).

A similar argument shows that such groups, and, more generally, almost direct products of residually (torsion-free) nilpotent groups are residually (torsion-free) nilpotent, see for instance Corollary 2 from V. Bardakov and P. Bellingeri, On residual properties of pure braid groups of closed surfaces, Comm. Algebra 37 (2009), no. 5, 1481–1490.