Version of 10-18-2001 BASIC COMMUTATORS AS RELATORS

Charles Sims, [7] has asked whether or not the lower central subgroup γn(F ) of a free group F coincides with the normal closure in F of the set of basic commutators of weight n. Here, we investigate variations of this question where we consider other varieties or other sets of commutators. We also give brief new proofs that this question has a positive answer in the previously known cases for weight n at most 4 and for the case where n = 5 and F is free with rank 2. We show that for n ≥ 4, γn(F ) is the normal closure in F of the set of basic commutators having weights n through 2n− 4, inclusive. This is a common generalization of Sims’ result for weight 4 and a result of Martin Ward, but both of these earlier results are used in the proof. 1: Introduction and Notation For a natural number r ≥ 2, F = Fr is the free group on the ordered alphabet X = {x1, x2, . . . , xr}. We write y for x−1yx and [y, x] for the commutator y−1x−1yx. If A and B are subsets of any group G, then A is the normal closure of A in G and [A,B] is the subgroup of G that is generated by the set of elements [a, b] where a ∈ A and b ∈ B. We write [z, x, y] for the commutator [[z, x], y]. The weight of a commutator c is denoted by wt(c). We follow Marshall Hall’s definition for basic commutators. See, for example, [2] or [6]. Fix an order on the alphabet X. (1) The basic commutators of weight one are the letters of X taken in this order. (2) Having defined and ordered the basic commutators of weight less than n, the basic commutators of weight n are all of the commutators [ci, cj ] which satisfy the conditions: (a) ci and cj are basic commutators with n = wt(ci) + wt(cj). (b) In the order that has been chosen for basic commutators of weight less than n, cj < ci. (c) If ci = [cs, ct] where cs and ct are basic commutators, then ct ≤ cj in the order that has been chosen for basic commutators of weight less than n. (3) The basic commutators of weight n follow all of the basic commutators of weight less than n in the order for the basic commutators of weight less than n+1, but the basic commutators of weight n may be ordered arbitrarily. While we may order the commutators of weight n arbitrarily, the choices that we make will have consequences for which commutators of higher weights are basic and which are not. For example, depending upon which order we use for the commutators [x2, x1] and [x3, x1] in weight 2, either [[x2, x1], [x3, x1]] or else [[x3, x1], [x2, x1]] 1991 Mathematics Subject Classification. 20F05, 20F12, 20F18.