Free Differential Calculus, IV. The Quotient Groups of the Lower Central Series

The quotient groups Qn(G) =GnGn+i of the lower central series G = G1 D G, D G, D * * of a finitely generated group G are finitely generated abelian groups. Our object is to develop an algorithm for the calculation of Qn from any given finite presentation of G. As a preliminary step, the special case of a free group X is considered. It is known [2], [7] that, for a free group X of rank q, the group Qn(X) is a free abelian group whose rank is the Witt number 0b,(q), and a basis for QJ(X) has been exhibited by M. Hall [42]. Our approach is somewhat different in that we construct, by means of the free differential calculus, a basis for the dual group Qn* = Hom [QnJ]. The corresponding dual basis of Q. is not the same as the Hall basis, although it bears a superficial resemblance to it. In the course of this construction we re-prove Witt's result [7] that the elements of Xn are just those for which the non-constant terms of the Magnus expansion are all of degree at least n, in short, that the lower central groups coincide with the dimension groups of Magnus [2]. Further, we derive a complete set of finite identities for the coefficients in the Magnus expansion of an element of X. The algorithm for Qn(G) is to be found in the last section. The authors wish to thank Julian Brody for his help in simplifying the arguments, and for selection of the example in ?4.