Superpositions of free Fox derivations

Abstract

Fox derivations are an effective tool for studying free groups and their group rings. Let Fr be a free group of finite rank r with basis {f1,..., fr}. For every i, the partial Fox derivations ∂/∂fi и ∂/∂fi-1 are defined on the group ring ℤ[Fr]. For k / 2, their superpositions Dfϵi = = ∂/∂fϵki о ... о ∂/∂fϵk1, ϵ = (ϵ1,..., ϵk) Є{±1}k, are not Fox derivations. In this paper, we study the properties of superpositions Dfϵi. It is shown that the restrictions of such superpositions to the commutant F′r are Fox derivations. As an application of the obtained results, it is established that for any rational subset R of F′r and any i there are parameters k and ϵ such that R is annihilated by Dfϵi.