Let $F$ be a free group with basis $X = \{ x_i\}_{i\in I}$ and $\mathbb{Z}F$ the integer group ring.
Differentiation or derivation, $D$, in this context is defined using a sort of nonsymmetric analogue of the Leibniz rule: it is an additive map $D:\mathbb{Z}F\to\mathbb{Z}F$ such that for all $u,v\in F$,
The Fox partial derivatives $\frac{\partial}{\partial x_i}$ are defined by the rules
extended to the products $u = y_1\ldots y_n$ where $y_i = x_k$ or $y_i=x_k^{-1}$ for some $k = k(i)$ by the formula
This then implies that
Notice that the summands on the right-hand side are “of different length”.
The lemma given in derivation on a group allows the following alternative form of the above definition to be given:
For each $x \in X$, let
be defined by
for $y \in X$,
for any words, $w_1,w_2 \in F$,
Then these uniquely determine the Fox derivative of $F$ with respect to $x$.
The Fox derivatives give a way of expanding any derivation (differentiation) defined on $F$. For every differentiation
(This is a finite sum since $u$ will only involve finitely many of the generators.)
In particular if $\epsilon:\mathbb{Z}F\to\mathbb{Z}$ is the augmentation map given by $\epsilon:x_i\mapsto 1$, then the differentiation $u\mapsto u-\epsilon(u) 1_F$ satisfies
hence it belongs to the left ideal in $\mathbb{Z}F$ which is generated by $(x_i-1)$.
This construction is important in combinatorial group theory, particularly in the study of free products of groups and the study of metabelian group?s.
Given any group $G$ with a presentation $\langle X; R\rangle = F/N$ such that $F=\langle X\rangle$ is the free group on the set of letters $X$ and $N$ the normal closure of the set of relations $R$, let $\bar{G}:=G/[G,G]$, let $\phi:F\to G$, $\bar\phi:F\to \bar{G}$ be the canonical projections; denote by the same letter their linearizations for group rings $\phi:\mathbb{Z}F\to \mathbb{Z}G$ and $\bar\phi:\mathbb{Z}F\to\mathbb{Z}\bar{G}$. The Jacobi matrix of the presentation is the matrix
and also the projected matrix $\bar{J}$ which is the image of $J$ as a matrix over $\mathbb{Z}\bar{G}$. The determinant ideal $D_i$ of order $i$ of the matrix $\bar{J}$ is the ideal of $\mathbb{Z}\bar{G}$ generated by all minors (= determinants of submatrices) of size $i\times i$ in $\bar{J}$. The sequence $D_1,D_2,\ldots$ is invariant (up to some technical details), that is does not depend on the presentation. In the case when $G=\pi(S)$ where $S$ is the complement of a knot, $\bar{G}$ is an infinite cyclic group. Let $t$ be its generator; then the highest nonzero determinant ideal (of $\bar{J}$) in $\mathbb{Z}\bar{G}$ is a principal ideal, hence it has a normalized (in the sense that the heighest coefficient is $1$) generator, which is a polynomial in $t$. This polynomial is an invariant of the knot, the Alexander polynomial of the knot.
The orginal articles include:
R. H. Fox, Free differential calculus I: Derivation in the free group ring, Annals Math. (2) 57, 547–560 (1953) doi:10.2307/1969736
R. H. Fox, Free differential calculus II: The Isomorphism Problem of Groups, Annals Math. (2) 59 196–210 (1954); III:Subgroups, Annals Math. (2) 64, 407–419; IV: 71, 408–422 (1960)
with a nice introduction in
see also
B. Chandler, W. Magnus, The history of combinatorial group theory: a case study in the history of ideas, Springer 1982
R. Lyndon, P. Schupp, Combinatorial group theory, Ch. II.3, Springer 1977(Russian transl. Mir, Moskva 1980)
and more recently
Valentino Zocca, Fox calculus, symplectic forms and moduli spaces, Trans. AMS__350__, 4, (1998) 1429-1466, pdf
Connections to double Poisson structures/brackets are discussed in
See also