Let $f : F \to G$ be a homomorphism from a free group $F$ to a group $G$. I heard that, in order to verify whether or not $f$ is one-to-one, it is possible to associate a Lie algebra $E_0^*(H)$ to any group $H$ using its derived series, and then to consider the associated map $$E_0^*(f) : E_0^*(F) \to E_0^*(G).$$
Unfortunately, I never really studied Lie algebras so the argument was obscur for me. Do you have a reference where I could find some explanation?
For any group $G$ denote by $G=G_1\supseteq G_2\supseteq G_3\supseteq \ldots$ the descending lower central series of $G$ defined inductively by $G_1 = G$ and $G_k = [G_{k−1},G_1]$ for $k\ge 2$. Then $$ gr(G)=\bigoplus_{i\ge 1} \frac{G_i}{G_{i+1}} $$ is the associated graded Lie ring, where the Lie bracket is induced by the group commutator. Let $$ \mathfrak{g}=gr(G)\otimes_{\mathbb{Z}}\mathbb{Q} $$ be the correspondig graded Lie algebra over $\mathbb{Q}$. For references and related generalizations see the answers here. There is also, among others, an article by Labute on this subject.