On the Magnus-smelkin Embedding

The generalization of the Magnus embedding [7] proved by Smelkin [9] may be stated as follows. Let L be a free group freely generated by the set x^iel) , and let R be a normal subgroup of L with G = L/R. If V is any variety of groups and n is the F-free group with free generating set the symbols [g,xj (geG, iel), then L/V(R) is embedded in the semidirect product II xi G (where the action of G on II is given by h • [g, x;] = [hg, x j , for h,geG). In addition to the considerable number of direct applications of the Magnus-Smelkin embedding that are now known, the Magnus embedding itself may be used to obtain representations of certain subgroups of the automorphism group of a free group, for example the Burau representation of Artin's Braid Group (see e.g. [1]). In the present paper we give an extension of the Magnus-Smelkin embedding to the situation where the free group L is replaced by an arbitrary group F. The embedding result is stated in Section 2. We then show, in Section 4, that this generalised embedding also gives rise to representations of automorphism groups, in the same way as does the Magnus representation. Our approach to these results may be summarized as follows. Let F be a group with normal subgroup R, and let G = F/R, where n is the natural map F->G with kernel R. Then there is a natural homomorphism rc, from the free product F * G to G (where 7̂ is induced by the identity map on G and by n on F). Let IT be the kernel of nv Since r i n G = l , and fn(f)~ belongs to II for each feF, it is clear that F*G can be described as the semidirect product II x G (where G acts on n by conjugation). The embedding result arises by factoring out suitable G-invariant normal subgroups of n from this product (the basic properties of n required for this are given in Theorem 1 below). The results on representations of automorphisms are then obtained from the following observation: let H be the subgroup of AutF consisting of those automorphisms <x of F such that a(R) = R. Each such a induces an automorphism a, of G, and hence an automorphism of F*G (namely a on F, a.x on G). The subgroup II of F*G is easily seen to be generated by all fn(f)~ (feF) and it follows that each aeH induces an automorphism of II. We thus obtain a homomorphism H-*k\xtT\ and hence a homomorphism J/-»Aut(Il/N), for N any characteristic subgroup of II.