# Free groups are residually of rank 2

Let $$F_X$$ denote the free group on the set $$X$$, and $$F_n$$ the free group of rank $$n$$.

I have read that any free group is residually $$F_2$$, and I was trying to understand this.

For any free group $$F$$, it is residually free of finite rank: given $$g \in F$$ with $$g \neq 1$$, let $$S$$ be a set of generators of $$F$$ that we can use to express $$g$$, then $$g$$ is not sent to the identity under the projection $$F \to F_S$$.

So the question reduces to why $$F_n$$ is residually $$F_2$$, for $$n \geq 2$$.

Edit: I realized that this is trivial from the embedding of $$F_n$$ in $$F_2$$, for any $$n \geq 1$$. So the interesting question is whether the morphism $$F_n \to F_2$$ can be chosen to be surjective. More explicitly: let $$n \geq 2$$, and $$1 \neq g \in F_n$$. Is there an epimorphism $$\phi : F_n \to F_2$$ such that $$\phi(g) \neq 1$$?

• @Shaun I am using the formal construction with words and concatenation. I do not remember in which paper I read it, but it was used to show that all free groups are residually finite (the residual finiteness of $F_2$ follows from it being embeddable in $GL_2(\mathbb{Z})$). Dec 27, 2018 at 18:21
• Related. Dec 27, 2018 at 19:08
• Hint: Every finite rank free group embeds in $F_2$. Dec 28, 2018 at 1:06
• One can prove more: Given a finite rank free group $F$ and a nontrivial element $g\in F$, there exists an epimorphism $\phi: F\to F_2$ such that $\phi(g)\ne 1$. But this is a more difficult result. Dec 28, 2018 at 17:31
• If you modify the question I will write this as an answer: It requires a bit of an argument. Dec 28, 2018 at 21:06

The argument I had in mind uses the "super-strong approximation property".

Every finitely generated free group $$F$$ embeds in $$SL(2, {\mathbb Z})$$ which is a 2-generated group. I will need two facts about $$SL(2, {\mathbb Z})$$, one of which is elementary and the other is hard:

a. For every finite subset $$A\subset SL(2, {\mathbb Z})$$, for all but finitely many primes $$p$$, the projection of $$A$$ to $$SL(2, {\mathbb Z}/p {\mathbb Z})$$ is 1-1.

b. If $$B\subset SL(2, {\mathbb Z})$$ consists of matrices generating a free subgroup $$G$$ of rank 2, then for all but finitely many primes $$p$$, the projection of $$B$$ to $$SL(2, {\mathbb Z}/p {\mathbb Z})$$ generates $$SL(2, {\mathbb Z}/p {\mathbb Z})$$. This is a nontrivial fact, see

R. Matthews, L. N. Vaserstein, B. Weisfeiler, "Congruence Properties of Zariski‐Dense Subgroups, I", Proc. London Math. Soc., Series 3, 48 (3) 1984, p. 514-532.

(They proved a much more general result, I am using only a special case needed here.)

I will take $$G$$ to be a rank 2 free factor of the free subgroup $$F< SL(2, {\mathbb Z})$$ and let $$A\subset F$$ be an arbitrary finite subset. By taking a suitable prime $$p$$, we get that:

i) The restriction of the projection $$\phi: SL(2, {\mathbb Z})\to SL(2, {\mathbb Z}/p {\mathbb Z})$$ to $$A$$ is 1-to-1.

ii) $$\phi(G)= SL(2, {\mathbb Z}/p {\mathbb Z})$$.

Since $$SL(2, {\mathbb Z}/p {\mathbb Z})$$ is a 2-generated group, there exists an epimorphism $$\eta: F_2\to SL(2, {\mathbb Z}/p {\mathbb Z}).$$ Since the group $$F$$ is free and $$G$$ is its free factor, there is a lift of the homomorphism $$\phi$$ to a homomorphism $$\psi: F\to F_2, \phi=\eta\circ \psi,$$ such that $$\psi(G)=F_2$$; hence, $$\psi$$ is an epimorphism. At the same time, the restriction of $$\psi$$ to $$A$$ is 1-to-1 since $$\phi$$ already has this property. Thus, we proved:

Theorem. For every free group $$F$$ of finite rank and for every finite subset $$A\subset F$$, there exists an epimorphism $$\psi: F\to F_2$$ whose restriction to $$A$$ is 1-to-1.

• Are you surprised this result is hard to prove? Because I am. As you've shown, it's enough to get a surjection $F\rightarrow G$, where $1<rk(G)<rk(F)$, and $g$ isn't in the kernel. I tried doing it for $G$ a finite $p$-group, knowing I could play with which prime $p$, but couldn't make it work. Dec 30, 2018 at 3:18

Yes, it's true: for every finite subset $$S$$ of a free group $$F$$, there exists a quotient $$F'$$ of $$F$$, such that $$F'$$ is free of rank 2, and such that $$S$$ is mapped injectively into $$F$$.

See for instance (d) p11 in Champetier, Guirardel, Limit groups as limits of free groups: compactifying the set of free groups. (arxiv link)