# Magnus' embedding theorem

I am looking for a (preferably modern) reference to the following old result of Magnus.

Let $F$ be a free group of finite rank and $$F_1 = [F,F], F_2 = [F_1,F_1], \dots , F_{n+1} = [F_n,F_n], \dots$$ Let $T_2$ be the group of all affine transformations $x \mapsto ax+b$ of the complex line. Here, both $a$ and $b$ are complex numbers and $a \ne 0$. Of course, one may view $T_2$ as the group of invertible upper-triangular $2\times 2$ complex matrices with $1$ at the lower right corner.

Theorem of Magnus (Crelle, 170 (1934)?): There exists an embedding of the quotient $F/F_2$ into $T_2$.

I also wonder if there are any generalizations of this result which deal with embeddings of quotients $F/F_n$ into groups of complex (upper-triangular) matrices.

• Magnus' theorem is not only that there exists an embedding, but that some given explicit homomorphism is an embedding (which is stronger and more useful).
– YCor
May 14, 2015 at 11:50
• @YCor I would appreciate a reference where such a homomorphism is explicitly described. May 14, 2015 at 17:07

Assuming $F$ non-abelian, there is no embedding of $F/\mathcal{D}^nF$ for $n\ge 3$ into a group of matrices over any field. (Here I write $\mathcal{D}G=[G,G]$, so $F_n=\mathcal{D}^nG$ in your notation.) One reason is that any solvable group of matrices is nilpotent-by-abelian-by-finite.

But $F/\mathcal{D}^nF$ is not nipotent-by-abelian-by-finite when $n\ge 3$. This is well-known, but here's a proof. Clearly it suffices to do the case when $F$ is free of rank 2. Indeed assume, by contradiction, otherwise. Then it follows that for some $k_0$, $F/\mathcal{D}^nF$ has a subgroup of index $k_0$ that is nilpotent-by-abelian; hence $F/\mathcal{D}^3F$ (being a quotient of $F/\mathcal{D}^nF$) has the same property. In turn, it follows that every 2-generated $3$-solvable group has a subgroup of index $\le k_0$ that is nilpotent-by-abelian.

Now in order to get a contradiction, consider three primes $p,q,r$, all $>k_0$, such that $q$ divides $r-1$, and fix a nontrivial action of the cyclic group $C_q$ on $C_r$. Consider the wreath product $G=(C_r\rtimes C_q)\rtimes C_p$. Let's show it's 2-generated. It clearly 3-generated: let $s,t,u$ be generators of $C_p$, $C_q$, $C_r$. If $g\in C_r\rtimes C_q$ and $k$ is integer, write $g_k=s^kgs^{-k}$ (so $g=g_0$). Then $G=\{s,t_0,u_0\}$. Now observe that $G=\{s,t_0u_1\}$ (use that $t_0$ and $u_1$- commute and have coprime orders), so $G$ is 2-generated. Also, the derived subgroup of $G$ contains $C_r^p$ (since $C_r$ is in the derived subgroup of $C_r\rtimes C_q$), and contains the element $t_Ot_1^{-1}$, which is a commutator of $s$ and $t_0$. In particular, the derived subgroup $\mathcal{D}G$ of $G$ contains an element of order $r$, say $u_0$ and an element of order $q$, say $t_0t_1^{-1}$, that do not commute; hence $\mathcal{D}G$ is not nilpotent; in other words, $G$ is not nilpotent-by-abelian.

Now turning back to the contradictory hypothesis, $G$ should have a nilpotent-by-abelian subgroup $H$ of index $\le k_0$. Since $G$ is not nilpotent-by-abelian, we have $H\neq G$. On the other hand, because $p,q,r>k_0$, $G$ has no proper index subgroup of index $\le k_0$. We get a contradiction, and hence $F/\mathcal{D}^nF$ is not nilpotent-by-abelian-by-finite (and nor is its profinite completion, by the way) and in particular $F/\mathcal{D}^nF$ is not isomorphic to a group of matrices over any field (or product of fields).

• Thanks! It seems that what you denote by $F_n$ is $F/F_n$ in my notation. May 14, 2015 at 16:01

I believe you are talking about the "Magnus Embedding", which has been studied by Fox (via the Free differential calculus - see his '53 paper). The most recent reference seems to be Kapovich-Drutu (Misha will probably give a better answer).