Representations of some relatively free groups in power series rings

Abstract

W. Magnus represents a free group in a formal power series ring with no relations. We obtain power series representations for certain relatively free groups by putting various relations on the set of variables of the power series. Among those we obtain power series representations for are F/Fm (the free, nilpotent groups) , F/F" (the free metabelian group) F/(F')3(F3)', F/(F')3(F4)', F/[F", F] (the free centre by metabelian group), F/[F",F,F] (the free centre by centre by metabelian group), and F/[F",F,F,F](F')3. In the process it is shown that F"/[F",F] is free abelian and an explicit basis is given. This basis is used to derive a basis for [F",F]/[F",F,F] and various other subgroups of the group's, for which we obtain power series representations, are shown to be free abelian. We prove that all these groups mentioned above are residually torsion free nilpotent using their power series representations. W. Magnus has also proved that the so-called dimension subgroups and the lower central factors of the free group coincide. In Chapter 5 we present analogues of this result of Magnus for the groups F/F", F/(F')3(F3)' and F/(F')3(F4)' and in the process, compute the structure of the lower central factors of these three groups. We conclude with a contribution to a problem of Fox on the determination of certain ideals in the group ring of the free group.