Theorem 3

For all $x \in X$, let $c(x)$ be the canonical image of $x$ in $F(X)/(F(X), F(X))$. Let $g$ be the graded Lie $\mathbf{Z}$-algebra associated with the filtration $(C^n F(X))_{n \geq 1}$ of $F(X)$ (§ 4, no. 6). The unique homomorphism of the free Lie $\mathbf{Z}$-algebra $L_\mathbf{Z}(X)$ into $g$ which extends $c$ is an isomorphism.

Loosely speaking, the graded Lie $\mathbf{Z}$-algebra associated with the free group $F(X)$ (with the lower central series) is the free Lie $\mathbf{Z}$-algebra $L_\mathbf{Z}(X)$.

We write $F(X) = F, \Gamma(X) = \Gamma, \hat{A}(X) = \hat{A}, \hat{A}_\mathbf{Z}(X) = \hat{A}_\mathbf{Z}, C^n F(X) = C^n, \Gamma_n = 1 + \hat{A}_n(X)$ and let $\alpha : L_\mathbf{Z}(X) \to g$ be the homomorphism introduced in the statement of Theorem 3.

(A) Preliminary reductions.

Let $\gamma$ denote the homomorphism of $F$ into $\Gamma$ defined by $\gamma(x) = 1 + x$ for $x \in X$. By Lemma 4 there exists an automorphism $\sigma$ of the algebra $\hat{A}$ compatible with the filtration on $\hat{A}$ and such that $\sigma(1 + x) = g(x)$ for all $x \in X$; then $\sigma(\Gamma_n) = \Gamma_n$ for all $n$. As the homomorphisms $g$ and $\sigma \circ \gamma$ of $F$ into $\Gamma$ coincide on $X, g = \sigma \circ \gamma$ and hence $g^{-1}(\Gamma_n) = \gamma^{-1}(\Gamma_n)$. Under the hypotheses of Theorem 2, $\mathbf{Z}$ can be identified with a subring of $K$; the Magnus algebra $\hat{A}_\mathbf{Z}$ is therefore identified with a subring of $\hat{A}$ and the filtration on $\hat{A}_\mathbf{Z}$ is induced by that on $\hat{A}$. As $\gamma$ maps $F$ into $\hat{A}_\mathbf{Z}$, we see that it suffices to prove Theorems 2 and 3 under the supplementary hypotheses $K = \mathbf{Z}, r = 0$ and hence $g = \gamma$, hypotheses which we shall henceforth make.

(B) Surjectivity of $\alpha$.

As $X$ generates the group $F = C^1$, the set $c(X)$ generates the $\mathbf{Z}$-module $g^1 = C^1/C^2$. But $g^1$ generates the Lie $\mathbf{Z}$-algebra $g$ (§ 4, no. 6, Proposition 5) and hence $c(X)$ generates $g$, which proves that $\alpha$ is surjective.

(C) We identify the graded algebra $\mathrm{gr}(\hat{A})$ with $A(X)$ under the canonical isomorphisms $A^n(X) \to \hat{A}_n/\hat{A}_{n+1}$. For every integer $n \geq 1$, we write $F^n = \gamma^{-1}(\Gamma_n)$; we know (§ 4, no. 5) that $(F^n)_{n \geq 1}$ is an integral central filtration on $F$. Let $g'$ denote the associated graded Lie $\mathbf{Z}$-algebra (§ 4, no. 4). Let $f$ be the Lie algebra homomorphism of $g'$ into $A(X)$ associated with $\gamma$ (§ 4, no. 5, Proposition 3). Now $C^n \subset F^n$ for every integer $n \geq 1$ (§ 4, no. 6, Proposition 4) and hence there is a canonical homomorphism $\varepsilon$ of $g = \bigoplus_{n \geq 1} C^n/C^{n+1}$ into $g' = \bigoplus_{n \geq 1} F^n/F^{n+1}$

We write $\beta = f \circ \varepsilon$; we give $\beta$ explicitly as follows: if $u$ is the class modulo $C^{n+1}$ of an element $w$ of $C^n$, then $\gamma(w) - 1$ is of order $\geq n$ in $\hat{A}$ and $\beta(u)$ is the homogeneous component of $\gamma(w) - 1$ of degree $n$. In particular,

(D) Proof of Theorems 2 and 3.

The Lie algebra homomorphism $\beta \circ \alpha : L_\mathbf{Z}(X) \to A(X)$ restricted to $X$ is the identity by (3) and hence is the canonical injection (§ 3, no. 1). Therefore $\alpha$ is injective and hence bijective by (B); this proves Theorem 3. As $\beta \circ \alpha = f \circ \varepsilon \circ \alpha$ is injective and $\alpha$ is bijective, $\varepsilon$ is injective. For all $n \geq 1$,

is injective and hence

C^1 = F = F^1; if C^n = F^n, then C^n \cap F^{n+1} = F^{n+1} whence C^{n+1} = F^{n+1} which proves Theorem 2 by induction on n \geqslant 1.