Finer Analogues of Milnor’s Link Groups

Let Gk denote the quotient of the fundamental group G of a link in S 3 by the product of the (k + 2)-nd lower cental series terms of the normal closures of meridians, k > 0. In contrast to a theorem of Waldhausen, we show that Gk is the finest (in certain sense) quotient of G, invariant under k-quasi-isotopy (which is a higher order analogue of link homotopy, studied in [MR1]) and at the same time Gk is not a complete invariant of k-quasi-isotopy, even if considered together with the finest (in certain sense) peripheral structure Pk . The invariants of k-quasiisotopy that cannot be extracted from (Gk ,Pk) include the generalized Sato–Levine invariant, Cochran’s derived invariant β, provided that k ≥ 3, and a series of μ̄invariants starting with μ̄(111112122) for k = 3. In fact, all μ̄-invariants where each index occurs at most k+1 times, except possibly for one index occuring k+2 times, can be extracted from (Gk,Pk), moreover vanishing of these μ̄-invariants implies that (Gk ,Pk) is the same as that of the unlink. We also prove that a group generated by two 3-Engel elements has lower central series length ≤ 5 (in fact we observe in §4 that groups generated by their (k+1)-Engel elements are related to the k-th Milnor’s link in the same way as groups where normal closures of generators are nilpotent of class k are related to (k− 1)-quasi-isotopically trivial links), record a lemma (3.13.iii) which is helpful for finding maximal linearly independent subsets of μ̄-invariants, and discuss multi-component analogues of the generalized Sato–Levine invariant.